On defining relations of the affine Lie superalgebras and their quantized universal enveloping superalgebras

Hiroyuki Yamane^{*}

11th March 1996

Introduction. In this paper, we give defining relations of the affine Lie superalgebras and defining relations of a super-version of the Drinfeld[D1]-Jimbo[J] affine quantized universal enveloping algebras. As a result, we can exactly define the affine quantized universal enveloping superalgebras with generators and relations. Moreover we give a Drinfeld’s realization of U_{h}(ŝl(m|n)^{(1)}).

For the Kac-Moody Lie algebra G, Gabber-Kac [GK] proved the Serre theorem which states that G is defined with the Chevalley generators H_{i}, E_{i}, F_{i} (1 ≤ i ≤ rankG) and relations

[H_{i}, H_{j}] = 0, [H_{i}, E_{j}] = (α_{i}, α_{j})E_{j}, [H_{i}, F_{j}] = −(α_{i}, α_{j})F_{j},
[E_{i}, F_{j}] = δ_{ij}H_{i},
ad(E_{i})^{1−a_{ij}}(E_{j}) = 0, ad(F_{i})^{1−a_{ij}}(F_{j}) = 0

where {α_{i}(1 ≤ i ≤ rankG)} are simple roots of G, ( , ) is an invariant form of G and (a_{ij}) is the Cartan matrix of G. We call these relations Serre’s relations.

Kac [K2] classified the finite dimensional simple Lie superalgebras, which are sl(m|n), osp(m|n), D(2, 1; x) (x ≠ 0, 1), F_{4}, G_{3}, P and Q. Van de Leur [VdL] classified the Kac-Moody Lie superalgebras G of finite growth, which are the finite dimensional simple Lie superalgebras and:

\begin{matrix}ŝl(m|n)^{(i)} (i = 1, 2, 4), ôsp(m|n)^{(i)} (i = 1, 2), \\ D(2, 1; x)^{(1)} (x ≠ 0, 1), F_{4}^{(1)}, G_{3}^{(1)}. \end{matrix}

In this paper we call complex infinite dimensional Kac-Moody Lie superalgebras of finite growth affine Lie superalgebras.

Our first result is to give a Serre theorem for the affine Lie superalgebra G, i.e., to give defining relations between the Chevalley generators H_{i}, E_{i}, F_{i}. We give the defining relations associated to each Cartan matrix of G. (In general, G does not have a unique Cartan matrix.) To do this, we use Weyl-group-type isomorphisms {L_{i}} between G. Let H be a Cartan subalgebra of G. We note that the Cartan matrix defined for H does not necessarily coincide with the one defined for L_{i}(H), though {L_{i}} are introduced as counterparts of the inner automorphisms \{exp(−adF_{i}) exp(−\frac{2}{(α_{i},α_{i})}adE_{i}) exp(−adF_{i}) (1 ≤ i ≤ rankG) \} of the Kac-Moody Lie algebra G. We introduce another Lie superalgebra Ḡ associated to each G. We define the Lie superalgebras Ḡ by a universal condition that {L_{i}} can be lifted to isomorphisms {L̄_{i}} between Ḡ. We directly calculate defining relations of Ḡ. In the case of the Kac-Moody Lie algebra G, defining relations of Ḡ are given by Serre’s relations. However, for G, we need other relations such as

*Dept. of Math., Osaka Univ., Toyonaka 560 Japan and DPMMS, Cambridge Univ., 16 Mill Lane, Cambridge CB2 1SB, UK

[[E_{i}, E_{j}], [E_{j}, E_{k}]] = 0 for (α_{i}, α_{k}) = (α_{j}, α_{j}) = 0, (α_{i}, α_{j}) = −(α_{j}, α_{k}) ≠ 0.

There is an epimorphism j : Ḡ → G satisfying j ∘ L_{i} = L̄_{i} ∘ j. In the case of the Kac-Moody Lie algebra, most of the proof of the Gabber-Kac theorem [GK] was to prove ker j = 0.

However, in the case of the affine Lie superalgebras G, Ḡ ≠ G if and only if G = ŝl(m|n)^{(i)} (i = 1, 2, 4) and m = n. (In the case of G = ŝl(m|m)^{(1)}, Ḡ = sl(m|m) ⊗ C[t, t^{−1}] ⊕ Cc ⊕ Cd and ker j = I ⊗ C[t, t^{−1}] where I is the unit matrix.) Nevertheless, we can also look for defining relations of G = ŝl(m|m)^{(i)} because we have concretely known ŝl(m|m)^{(i)}.

Our second result is to give relations of quantized universal enveloping superalgebras U_{h}(G) such that, after h → 0, the relations become the defining relations of Ḡ obtained as our first result. Here U_{h}(G) is an h-adic topological C[[h]]-Hopf superalgebra introduced in [Y1]. In [Y1], we showed an existence of a non-degenerate symmetric bilinear form defined on a Borel part of U_{h}(G).

Applying the Drinfeld’s quantum double construction to U_{h}(G) by using the bilinear form, we can see that U_{h}(G) is topologically free and that the universal R-matrix of U_{h}(G) exists.

Since U_{0}(G) = U_{h}(G)/hU_{h}(G) is a cocommutative Hopf C-superalgebra, applying Milnor-Moor theorem [MM] to U_{0}(G), we see that U_{0}(G) is a universal enveloping superalgebra U(G_{0}) of the Lie superalgebra G_{0} = P(U_{0}(G)) of primitive elements of U_{0}(G). By definition of G as the Kac-Moody Lie superalgebra, G must be a quotient of G_{0}. On the other hand, we see that G_{0} is a quotient of Ḡ by our second result. Hence, if G = Ḡ, we see that U_{0}(G) coincides with U(G) and that our relations must be defining relations of U_{h}(G) by the topologically freedom of U_{h}(G).

Finally, we calculate relations of U_{h}(ŝl(m|m)^{(1)}) which become ones generating ker j after h → 0, while showing Drinfeld’s realization of U_{h}(ŝl(m|n)^{(1)}) for general m, n. Gathering up the relations and the ones obtained as our second result, we get defining relations of U_{h}(ŝl(m|m)^{(1)}). To do these, we introduce a Braid group action on U_{h}(ŝl(m|n)^{(1)}) which become the action on ŝl(m|n)^{(1)} defined by {L_{i}} after h → 0, and follow Beck’s argument [B]. We won’t consider U_{h}(ŝl(m|m)^{(2)}) or U_{h}(ŝl(m|m)^{(4)}).

Results in this paper have already been announced in [Y2]. The same results for the finite dimensional A − G type simple Lie superalgebras have already been given in [Y1].

1 Preliminary

1.1. In §1, we mainly refer to [K1-2] and [VdL].

Let G be a C-vector space with a direct sum decomposition G = G(0) ⊕ G(1). For X ∈ G, p(X) ∈ {0, 1} means that X ∈ G(p(X)). We call p(X) the parity of X. A Lie superalgebra G is defined with the bilinear map [ , ] : G × G → G such that

[X, Y] = −(−1)^{p(X)p(Y)}[Y, X],
[X, [Y, Z]] = [[X, Y], Z] + (−1)^{p(X)p(Y)}[Y, [X, Z]].

If a bilinear form ( | ) : G × G → C satisfies (X|Y) = (−1)^{p(X)p(Y)}(Y |X) and ([X, Y]|Z) = (X|[Y, Z]), then we call it an invariant form.

A Lie superalgebra Ĝ = G ⊗_{C} C[t, t^{−1}] ⊕ Cc ⊕ Cd is defined by

[X ⊗ t^{m} + a_{1}c + b_{1}d, Y ⊗ t^{n} + a_{2}c + b_{2}d]
= [X, Y] ⊗ t^{m+n} + mδ_{m+n,0}(X|Y)c + b_{1}nY ⊗ t^{n} − b_{2}mX ⊗ t^{m}.

where Ĝ(0) = G(0) ⊗_{C} C[t, t^{−1}] ⊕ Cc ⊕ Cd and Ĝ(1) = G(1) ⊗_{C} C[t, t^{−1}].

Let γ : G → G be an automorphism of finite order r (i.e. γ([X, Y]) = [γ(X), γ(Y)]). Put

G_{n}^{γ} = \{X ∈ G|γ(X) = (exp\frac{2π\sqrt{−1}}{n})X\} (0 ≤ n < r).

Then G_{0}^{γ} is a subalgebra of G and G_{i}^{γ} (1 ≤ i ≤ r −1) is the G_{0}^{γ}-module. We can define a subalgebra Ĝ^{(γ)} by

Ĝ^{(γ)} = ⨁_{n=0}^{r−1}( ⨁_{m∈Z} G_{n}^{γ} ⊗ t^{mr+n}) ⊕ Cc ⊕ Cd .

Obviously Ĝ^{(1)} = Ĝ.

1.2. Here we introduce a definition of the Kac-Moody Lie superalgebra in an abstract manner similar to the abstract definition of the Kac-Moody Lie algebra given in [K1;1.3]. Let E be a finite dimensional C vector space with a nondegenerate symmetric bilinear form ( , ) : E × E → C. Let Π = {α_{0}, ..., α_{1}} be a finite linearly independent subset of E. We call an element α_{i} ∈ Π the simple root. Let p : Π → Z/2Z be a function. We call p the parity function. Put H = E^{∗}. We call H the Cartan subalgebra. We identify an element γ ∈ E with H_{γ} ∈ H satisfying δ(H_{γ}) = (δ, γ) (δ ∈ E). For a datum (E, Π, p), we define a Lie superalgebra G̃ = G̃(E, Π, p) with the generators E_{i}, F_{i} (0 ≤ i ≤ n), H ∈ H, the parities p(E_{i}) = p(F_{i}) = p(α_{i}), p(H) = 0 and the relations:

[H, H′] = 0 (H, H′ ∈ H),  (1.2.1)
[H, E_{i}] = α_{i}(H)E_{i}, [H, F_{i}] = −α_{i}(H)F_{i},  (1.2.2)
[E_{i}, F_{j}] = δ_{ij}H_{α_{i}}.  (1.2.3)

Then we have the triangular decomposition G̃ = Ñ^{+}⊕H⊕Ñ^{−}. Here Ñ^{+} (resp. Ñ^{−}) is the free superalgebra with generators E_{i} (resp. F_{i}). Define the quotient Lie superalgebra G = G(E, Π, p) of G̃ = G̃(E, Π, p) by

G(E, Π, p) = G̃(E, Π, p)/r .

where r = r(E, Π, p) is the ideal which is maximal of the ideals r_{1} satisfying r_{1} ∩ H = 0. We call G = G(E, Π, p) the Kac-Moody Lie superalgebra. We have the triangular decomposition

G = N^{+} ⊕ H ⊕ N^{−}

where N^{+} and N^{−} are the subalgebras of G generated by E_{i} and F_{i} respectively. Let r_{±} = r∩Ñ^{±}. Then we have r = r_{−} ⊕ r_{+} and N^{±} = Ñ^{±}/r_{±}. We also have Ñ^{+} ≅ Ñ^{−} (E_{i} ↔ F_{i}). Let G = H ⊕ (⨁_{α∈E} G_{α}) be the root space decomposition where G_{α} = {X ∈ G|[H, X] = α(H)X (H ∈ H)}. Let r_{α} = r ∩ G_{α}. We put Φ = Φ(E, Π, p) = {α ∈ E|G_{α} ≠ 0}. Let P_{+} = Z_{+}α_{0} ⊕ ⋯ ⊕ Z_{+}α_{n} and Φ_{+} = Φ ∩ P_{+}, Φ_{−} = −Φ_{+}. Then we have Φ = Φ_{+} ∪ Φ_{−}. Clearly, we have r = ⨁_{α∈Φ} r_{α}.

For β, α ∈ P_{+}, we say β ≤ α if α − β ∈ P_{+}. Let r_{+,≤α} be the ideal of Ñ^{+} generated by r_{β} with β ≤ α. Then r_{+} = ∪_{γ∈P_{+}}r_{+,≤γ}. By the same argument in [K1], we have: Proposition 1.2.1. For (E, Π, p), let ρ ∈ E be an element such that (ρ, α_{i}) = (α_{i}, α_{i})/2. If α ∈ P_{+} satisfies (α, α) ≠ 2(ρ, α), then r_{α} is included in the ideal of Ñ^{+} generated by r_{β} such that β ≥ α. In particular,

r_{+} = ⋃_{γ∈P_{+}, (γ,γ)=2(ρ,γ)} r_{+,≤γ} .  (1.2.4)

Lemma 1.2.1. For α_{i} ∈ Π, we have dim G_{α_{i}} = 1. If p(α_{i}) = 1 and (α_{i}, α_{i}) ≠ 0, then dim G_{2α_{i}} = 1. Proof. By [E_{i}.F_{i}] = H_{α_{i}} ≠ 0, dim G_{α_{i}} ≠ 0. Hence dim G_{α_{i}} = 1. Similarly we can get a proof of the latter half.

Q.E.D.

1.3. Here we introduce the Dynkin diagram Γ for a datum (E, Π, p). We first prepare the threetype dots:

,  , [figure] [figure]  .

We call those the white dot, the gray dot and the black dot respectively. To the i-th simple root α_{i}, we put the corresponding i-th dot defined such that:

[figure: if (α_{i}, α_{i}) ≠ 0 and p(α_{i}) = 0 ,; if (α_{i}, α_{i}) = 0 and p(α_{i}) = 1 ,; if (α_{i}, α_{i}) ≠ 0 and p(α_{i}) = 1 .]

The dot [figure] stands for ^{}or [figure] . The dot [figure] stands for ^{}or ^{}. We write a |(α_{i}, α_{j})|-line between the i-th dot and the j-th dot or write as follows:

·i (α_{i}, α_{j})  · j

Moreover we add a pile pointing to the smaller of |(α_{i}, α_{i})| and |(α_{j}, α_{j})|. If |(α_{i}, α_{i})| = 0 or

|(α_{j}, α_{j})| = 0, then we sometimes omit the pile. The semilines [figure: ⋮; ⋮] stands for [figure] or [figure] .

If (α_{i}, α_{j}) ≠ 0, (α_{i}, α_{k}) ≠ 0 and \frac{|(α_{i}, α_{j})|}{(α_{i}, α_{j})} · \frac{|(α_{i}, α_{k})|}{(α_{i}, α_{k})} = 1, then we put ∗ in a sector enclosed by

an edge between i-th and j-th dots and an edge between i-th and k-th dots. Namely we describe

the situation as [figure: ×; i; ×; j; ×; k; ∗] . However we sometimes omit ∗.

1.4. We have already known: Theorem 1.4.1[K2].The Kac-Moody Lie superalgebra G(E, Π, p) is finite dimensional as a Cvector space if and only if the datum is one of the following Dynkin diagram. (In any diagram, there is an only one dot whose parity is odd.)

Diagram 1.4.1

[figure: A_{N−1}; ⋯; ×; ⋯]

[figure: B_{N}; ⋯; ×; ⋯; ⟹]

[figure: B_{N}; ⋯; ⟹]

[figure: C_{N}; ×; ⋯; ⟸]

[figure: D_{N}; ⋯; ×; ⋯; ∗]

[figure: D(2; 1, x); ×; −1; x; (x ≠ 0, 1)]

[figure: F_{4}; ⟹; ×]
[figure: G_{3}; ×; <]

1.5. In 1.5-13, we give a concrete form of finite dimensional G(E_{0}, Π_{0}, p_{0}) of (E_{0}, Π_{0}, p_{0}) of ABCD-type, which is given by using matrices.

Let Ẽ_{0} is an Ñ-dimensional C-vector space with a nondegenerate symmetric bilinear form ( , ) : Ẽ_{0} × Ẽ_{0} → C. Let e ∈ {±1}. Let ε̄_{i} (1 ≤ n ≤ N) be the orthogonal basis of Ẽ_{0} satisfying

(ε̄_{i}, ε̄_{j}) = e · δ_{ij} · (−1)^{p̃(i)} .

where p̃(i) is 0 or 1. Let d̄_{i} = (ε̄_{i}, ε̄_{i}). Let gl(Ẽ_{0}, e) = gl(Ẽ_{0}) be the C-linear space of Ñ × Ñ - matrices. Put E_{ij} = (δ_{xi}, δ_{yi})_{1≤x, y≤Ñ} ∈ gl(Ẽ_{0}) (1 ≤ i, j ≤ Ñ). We regard gl(Ẽ_{0}, e) as a superspace with a parity p defined by p(E_{ij}) = (−1)^{(p̃(i)+p̃(j))}. Then gl(Ẽ_{0}, e) can be regarded as a Lie superalgebra defined by

[X, Y] = XY − (−1)^{p(X)p(Y)}Y X .

Define a C-linear map str : gl(E_{0}, e) → C by str(E_{ij}) = δ_{ij} · d̄_{i}. Let sl(E_{0}, e) denote the subalgebra of gl(E_{0}, e) of the elements X ∈ gl(E_{0}, e) satisfying str(X) = 0. Let G(E_{0}, Π_{0}, p_{0}) be a finite dimensional Kac-Moody Lie superalgebra. Let γ : G(E_{0}, Π_{0}, p_{0}) → G(E_{0}, Π_{0}, p_{0}) be an automorphism of finite order r. In 1.6-13, We give a concrete form of an affine ABCD-type superalgebra G(E, Π, p) arising from Ĝ(E_{0}, Π_{0}, p_{0})^{(γ)}. Let (E_{0}^{γ}, Π_{0}^{γ}, p_{0}^{γ}) be the datum of G(E_{0}, Π_{0}, p_{0})_{0}^{γ}, i.e., G(E_{0}^{γ}, Π_{0}^{γ}, p_{0}^{γ}) = G(E_{0}, Π_{0}, p_{0})_{0}^{γ}. For γ, we define the datum (E, Π, p) = (E_{0}, Π_{0}, p_{0})^{(γ)} of affine type as follows:

(i) E = E_{0}^{γ} ⊕ Cδ ⊕ CΛ_{0}

where (x, δ) = (x, Λ_{0}) = 0 (x ∈ E_{0}^{γ}), (δ, δ) = (Λ_{0}, Λ_{0}) = 0 and (δ, Λ_{0}) = 1.

(ii) For the lowest weight ψ of G(E_{0}, Π_{0}, p_{0})_{1}^{γ}, let α_{0} = δ + ψ and Π = {α_{0}} ∪ Π_{0}^{γ}.

(iii) Let e_{ψ} ∈ G(E_{0}, Π_{0}, p_{0})_{1}^{γ} be a weight vector of ψ. We define the parity p(α_{0}) of α_{0} by the parity of e_{ψ} in G(E_{0}, Π_{0}, p_{0}). We also define p(α_{i}) = p_{0}^{γ}(α_{i}) (α_{i} ∈ Π_{0}^{γ})

1.6. Here we put N = Ñ, E_{0} = Ẽ_{0}, n = N − 1, N ≥ 2 and e = 1. Let (E_{0}, Π_{0}, p_{0}) be the datum whose Dynkin diagram is:

Diagram 1.6.1

[figure: A; 1; ε̄_{1} − ε̄_{2}; 2; ε̄_{2} − ε̄_{3}; …; N − 1; ε̄_{N−1} − ε̄_{N}]

Then we can identify G(E_{0}, Π_{0}, p_{0}) with gl(E_{0}, e) where Hε̄_{j} = d̄_{j}E_{jj}, E_{i} = E_{ii+1}, F_{i} = d̄_{i}E_{i+1i} . We also note that H_{α_{i}} = d̄_{i}E_{ii} − d̄_{i+1}E_{i+1i+1}.

We also note the lowest root of gl(E_{0}, e) is θ = ε̄_{N} − ε̄_{1}. Lowest and highest root vectors E_{θ}, F_{θ} (∈ gl(E_{0}, e)) satisfying [E_{θ}, F_{θ}] = H_{θ}(= d̄_{N}E_{NN} − d̄_{1}E_{11}) are given by

F_{θ} = d̄_{N}E_{1N}, E_{θ} = E_{N1}.

Proposition 1.6.1. If ∑_{i=1}^{N} d̄_{i} ≠ 0, then sl(E_{0}, e) is the simple Lie superalgebra. If ∑_{i=1}^{N} d̄_{i} = 0, then the quotient sl(E_{0}, e)/I is the simple Lie superalgebra where

I = C · ∑_{i=1}^{N−1}((∑_{j=1}^{i} d̄_{j}) · H_{α_{i}}) = C· ∑_{i=1}^{N} E_{ii}.

Proposition 1.6.2. Assume N ≥ 3 if p(α_{0}) = 1. Let (E_{0}, Π_{0}, p_{0}) be the datum of Diagram 1.6.1 and (E, Π, p) = (E_{0}, Π_{0}, p_{0})^{(I)}. Let ¯ Ḡ(E, Π, p) = ŝl(E_{0}, e) + E_{0} (⊂ ĝl(E_{0}, e)). There is an epimorphism j : ¯ Ḡ(E, Π, p) → G(E, Π, p) defined by letting j(H⊗1+ac+bd) = H+aH_{δ}+bH_{Λ_{0}} (H ∈ H) and j(E_{θ} ⊗ t) = E_{0}, j(F_{θ} ⊗ t^{−1}) = F_{0}.

If ∑_{i=1}^{N} d̄_{i} ≠ 0, then j is an isomorphism. If ∑_{i=1}^{N} d̄_{i} = 0, then

ker j = ⊕_{i≠0}I ⊗ t^{i}.

Example 1.6.1. The Dynkin diagram of (E, Π, p) in Proposition 1.6.2 is:

Diagram 1.6.2. (N ≥ 2)

[figure: (AA); 1; ε̄_{1} − ε̄_{2}; 2; ε̄_{2} − ε̄_{3}; …; N − 1; ε̄_{N−1} − ε̄_{N}; 0; δ − ε̄_{1} + ε̄_{N}]

[figure: 0; 1]
(∑_{i=0}^{N−1} p(α_{i}) ≡ 0).

1.7. Here we assume

Ñ = \{ \begin{matrix}2N + 1 & if Ñ is odd, \\ 2N & if Ñ is even. \end{matrix}

We assume Ñ ≥ 3. Let i′ = Ñ + 1 − i. We also assume

p̃(i) = p̃(i′) (1 ≤ i ≤ Ñ)

and

p̃(N + 1) = 0 if Ñ is odd.

Let g_{i} (1 ≤ i ≤ Ñ) be such that g_{i} ∈ {±1} and g_{i}g_{i′} = (−1)^{p̃(i)}. We assume that g_{N+1} = 1 if Ñ is odd.

Let Ω be an automorphism of gl(Ẽ_{0}, e) of order 2 defined by:

Ω(X)_{ij} = −(−1)^{(p̃(i)p̃(j)+p̃(j))}g_{i}g_{j}X_{j′i′} .

Denote sl(Ẽ_{0}, e)_{0}^{Ω} by osp(Ẽ_{0}, e). We can identify osp(Ẽ_{0}, e) with a finite dimensional Kac-Moody Lie superalgebra G(E_{0}, Π_{0}, p_{0}) of a datum (E_{0}, Π_{0}, p_{0}).

Here we note n = N. In 1.8-11 we will give:

(1) the datum (E_{0}, Π_{0}, p_{0}).

(2) the lowest root θ of sl(Ẽ_{0}, e)_{0}^{Ω}, a lowest root vector E_{θ} and a highest root vector F_{θ} such that [E_{θ}, F_{θ}] = H_{θ}.

(3) the lowest weight ψ of sl(Ẽ_{0}, e)_{1}^{Ω}, a lowest weight vector E_{ψ} and a highest weight vector F_{ψ} such that [E_{ψ}, F_{ψ}] = H_{ψ}.

(4) G(E, Π, p) and ¯ Ḡ(E, Π, p) arising from ôsp(Ẽ_{0}, e)^{(I)} and ŝl(Ẽ_{0}, e)^{(Ω)}.

1.8. B-type. If Ñ is odd, then the Dynkin diagram of (E_{0}, Π_{0}, p_{0}) is:

Diagram 1.8.1. [figure: B; 1; ε̄_{1} − ε̄_{2}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; ε̄_{N}; .]

Here Hε̄_{j} = d̄_{j}(E_{jj} − E_{j′j′}) (1 ≤ j ≤ N),

From: Hiroyuki Yamane ¡H.Yamane@pmms.cam.ac.uk¿ Subject: Hiroyuki To: H.yamane@pmms.cam.ac.uk Date: Tue, 27 Aug 1996 18:09:53 +0100 (BST) X-Mailer: ELM [version 2.4 PL25] Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Status: O

E_{i} = E_{ii+1}−(−1)^{p̃(i)p̃(i+1)}(−1)^{p̃(i+1)}g_{i}g_{i+1}E_{(i+1)′i′} (1 ≤ i ≤ N −1), E_{N} = E_{NN+1}−g_{N}E_{(N+1)′N′}, F_{i} = e · {(−1)^{p̃(i)}E_{i+1i} − (−1)^{p̃(i)p̃(i+1)}g_{i}g_{i+1}E_{i′(i+1)′}} (1 ≤ i ≤ N − 1), F_{N} = e · {(−1)^{p̃(N)}E_{N+1N} − g_{N}E_{N′(N+1)′}} .

Moreover we have: If p(α_{1}) + ⋯ + p(α_{N−1}) ≡ 0, then

θ = −ε̄_{1} − ε̄_{2}, F_{θ} = e · {(−1)^{p̃(1)}E_{21′} − (−1)^{p̃(1)p̃(2)}g_{2}g_{1′}E_{12′}},
E_{θ} = E_{1′2} − (−1)^{p̃(1)p̃(2)}(−1)^{p̃(2)}g_{1′}g_{2}E_{2′1},
ψ = −2ε̄_{1}, F_{ψ} = e · 2(−1)^{p̃(1)}E_{11′}, E_{ψ} = E_{1′1}.

If p(α_{1}) + ⋯ + p(α_{N−1}) ≡ 1, then

θ = −2ε̄_{1}, F_{θ} = e · 2(−1)^{p̃(1)}E_{11′}, E_{θ} = E_{1′1}.
ψ = −ε̄_{1} − ε̄_{2}, F_{ψ} = e · {(−1)^{p̃(1)}E_{21′} + (−1)^{p̃(1)p̃(2)}g_{2}g_{1′}E_{12′}},
E_{ψ} = E_{1′2} + (−1)^{p̃(1)p̃(2)}(−1)^{p̃(2)}g_{1′}g_{2}E_{2′1},

1.9. C-type. If Ñ is even and p̃(N) = 1, e = −d̄_{N}, then the Dynkin diagram of (E_{0}, Π_{0}, p_{0}) is:

Diagram 1.9.1. [figure: C; 1; ε̄_{1} − ε̄_{2}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; 2ε̄_{N}; .]

Here Hε̄_{j} = d̄_{j}(E_{jj} − E_{j′j′}) (1 ≤ j ≤ N), E_{i} = E_{ii+1} − (−1)^{(p̃(i)p̃(i+1)+p̃(i+1))}g_{i}g_{i+1}E_{(i+1)′i′} (1 ≤ i ≤ N − 1), E_{N} = E_{NN′}, F_{i} = e · {(−1)^{p̃(i)}E_{i+1i} − (−1)^{p̃(i)p̃(i+1)}g_{i}g_{i+1}E_{i′(i+1)′}} (1 ≤ i ≤ N − 1), F_{N} = e · 2(−1)^{p̃(N)}E_{N′N} .

Moreover we have: If p(α_{1}) + ⋯ + p(α_{N−1}) ≡ 1, then

θ = −ε̄_{1} − ε̄_{2}, F_{θ} = e · {(−1)^{p̃(1)}E_{21′} − (−1)^{p̃(1)p̃(2)}g_{2}g_{1′}E_{12′}},
E_{θ} = E_{1′2} − (−1)^{p̃(1)p̃(2)}(−1)^{p̃(2)}g_{1′}g_{2}E_{2′1},
ψ = −2ε̄_{1}, F_{ψ} = e · 2(−1)^{p̃(1)}E_{11′}, E_{ψ} = E_{1′1}.

If p(α_{1}) + ⋯ + p(α_{N−1}) ≡ 0, then

θ = −2ε̄_{1}, F_{θ} = e · 2(−1)^{p̃(1)}E_{11′}, E_{θ} = E_{1′1},
ψ = −ε̄_{1} − ε̄_{2}, F_{ψ} = e · {(−1)^{p̃(1)}E_{21′} + (−1)^{p̃(1)p̃(2)}g_{2}g_{1′}E_{12′}},
E_{ψ} = E_{1′2} + (−1)^{p̃(1)p̃(2)}(−1)^{p̃(2)}g_{1′}g_{2}E_{2′1}.

1.10. D-type. If Ñ is even and p̃(N) = 0, e = d̄_{N}, then the Dynkin diagram of (E_{0}, Π_{0}, p_{0}) is:

Diagram 1.10.1. [figure: D; 1; ε̄_{1} − ε̄_{2}; …; N − 2; ε̄_{N−2} − ε̄_{N−1}; ε̄_{N−1} − ε̄_{N}; N − 1; ε̄_{N−1} + ε̄_{N}; N; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; .]

Here Hε̄_{j} = d̄_{j}(E_{jj} − E_{j′j′}) (1 ≤ j ≤ N), E_{i} = E_{ii+1} − (−1)^{(p̃(i)p̃(i+1)+p̃(i+1))}g_{i}g_{i+1}E_{(i+1)′i′} (1 ≤ i ≤ N − 1), E_{N} = E_{N−1N′} − g_{N−1}g_{N′}E_{N(N−1)′}, F_{i} = e·{(−1)^{p̃(i)}E_{i+1i}−(−1)^{p̃(i)p̃(i+1)}g_{i}g_{i+1}E_{i′(i+1)′}} (1 ≤ i ≤ N−1), F_{N} = e·{(−1)^{p̃(N−1)}E_{N′N−1}− g_{(N−1)′}g_{N′}E_{(N−1)′N}} .

Moreover we have: If p(α_{1}) + ⋯ + p(α_{N−1}) ≡ 0, then

θ = −ε̄_{1} − ε̄_{2}, F_{θ} = e · {(−1)^{p̃(1)}E_{21′} − (−1)^{p̃(1)p̃(2)}g_{2}g_{1′}E_{12′}},
E_{θ} = E_{1′2} − (−1)^{p̃(1)p̃(2)}(−1)^{p̃(2)}g_{1′}g_{2}E_{2′1},
ψ = −2ε̄_{1}, F_{ψ} = e · 2(−1)^{p̃(1)}E_{11′}, E_{ψ} = E_{1′1}.

If p(α_{1}) + ⋯ + p(α_{N−1}) ≡ 1, then

θ = −2ε̄_{1}, F_{θ} = e · 2(−1)^{p̃(1)}E_{11′}, E_{θ} = E_{1′1}.
ψ = −ε̄_{1} − ε̄_{2}, F_{ψ} = e · {(−1)^{p̃(1)}E_{21′} + (−1)^{p̃(1)p̃(2)}g_{2}g_{1′}E_{12′}},
E_{ψ} = E_{1′2} + (−1)^{p̃(1)p̃(2)}(−1)^{p̃(2)}g_{1′}g_{2}E_{2′1},

1.11. Proposition 1.11.1. osp(Ẽ_{0}, e) is the simple Lie superalgebra.

Proposition 1.11.2. Let (E_{0}, Π_{0}, p_{0}) be the datum of osp(Ẽ_{0}, e) and (E, Π, p) = (E_{0}, Π_{0}, p_{0})^{(I)}. Put ¯ Ḡ(E, Π, p) = ôsp(Ẽ_{0}, e)^{(I)}. There is an isomorphism: j : ¯ Ḡ(E, Π, p) → G(E, Π, p) defined by letting j(H ⊗ 1 + ac + bd) = H + aH_{δ} + bH_{Λ_{0}} (H ∈ H) and j(E_{θ} ⊗ t) = E_{0}, j(F_{θ} ⊗ t^{−1}) = F_{0}.

Proposition 1.11.3. If ∑Ñ ; i=1(−1)^{p̃(i)} ≠ 0, then sl(Ẽ_{0}, e)_{1}^{Ω} is the simple osp(Ẽ_{0}, e)-module. If (−1)Ñ = 1 and ∑_{i=1}^{N} d̄_{i} = 0, then the quotient sl(Ẽ_{0}, e)_{1}^{Ω}/I is the simple osp(Ẽ_{0}, e)-module where

I = C· ∑_{i=1}^{N}(E_{ii} + E_{i′i′}) .

Proposition 1.11.4. Assume N ≥ 3 if α_{N} = 2ε̄_{N} and p(α_{1}) = 1. Let (E_{0}, Π_{0}, p_{0}) be the datum of sl(Ẽ_{0}, e) and (E, Π, p) = (E_{0}, Π_{0}, p_{0})^{(Ω)}. Put ¯ Ḡ(E, Π, p) = ŝl(Ẽ_{0}, e)^{(Ω)}. There is an epimorphism: j : ŝl(Ẽ_{0}, e)^{(2)} → G(E, Π, p) defined by letting j(H ⊗ 1 + ac + bd) = H + aH_{δ} + bH_{Λ_{0}} (H ∈ H) and j(E_{ψ} ⊗ t) = E_{0}, j(F_{ψ} ⊗ t^{−1}) = F_{0}.

If ∑_{i=1}^{N} d̄_{i} ≠ 0, then j is an isomorphism. If (−1)Ñ = 1 and ∑_{i=1}^{N} d̄_{i} = 0, then

ker j = ⊕_{i}I ⊗ t^{2i−1}.

.

Example 1.11.1. The Dynkin diagrams of (E, Π, p) in Proposition 1.11.2 (resp. Proposition 1.11.4) are:

Diagram 1.11.1 (N ≥ 2)

[figure: (DB); δ − ε̄_{1} − ε̄_{2}; 0; ε̄_{1} − ε̄_{2}; 1; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; 2; ε̄_{2} − ε̄_{3}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; ε̄_{N}]

[figure: 0; 1; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; 2]
(∑_{i=1}^{N} p(α_{i}) ≡ 0 (resp. 1))

Diagram 1.11.2 (N ≥ 1)

[figure: (CB); 0; δ − 2ε̄_{1}; 1; ε̄_{1} − ε̄_{2}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; ε̄_{N}]

[figure: 0; 1]
(∑_{i=1}^{N} p(α_{i}) ≡ 1 (resp. 0))

Diagram 1.11.3 (N ≥ 3)

[figure: (DC); δ − ε̄_{1} − ε̄_{2}; 0; ε̄_{1} − ε̄_{2}; 1; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; 2; ε̄_{2} − ε̄_{3}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; 2ε̄_{N}]
(∑_{i=1}^{N−1} p(α_{i}) ≡ 1 (resp. 0))

Diagram 1.11.4 (N ≥ 3)

[figure: (CC); 0; δ − 2ε̄_{1}; 1; ε̄_{1} − ε̄_{2}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; 2ε̄_{N}]
(∑_{i=1}^{N−1} p(α_{i}) ≡ 0 (resp. 1))

Diagram 1.11.5 (N ≥ 3)

[figure: (DD); δ − ε̄_{1} − ε̄_{2}; 0; ε̄_{1} − ε̄_{2}; 1; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮; 2; ε̄_{2} − ε̄_{3}; …; N − 2; ε̄_{N−2} − ε̄_{N−1}; ε̄_{N−1} − ε̄_{N}; N − 1; ε̄_{N−1} + ε̄_{N}; N; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮]

[figure: 0; 2; 1; 3]

(∑_{i=1}^{N−1} p(α_{i}) ≡ 0 (resp. 1))

Diagram 1.11.6 (N ≥ 3)

[figure: (CD); 0; δ − 2ε̄_{1}; 1; ε̄_{1} − ε̄_{2}; …; N − 2; ε̄_{N−2} − ε̄_{N−1}; ε̄_{N−1} − ε̄_{N}; N − 1; ε̄_{N−1} + ε̄_{N}; N; ⋮; ⋮; ⋮; ⋮; ⋮; ⋮]

(∑_{i=1}^{N−1} p(α_{i}) ≡ 1 (resp. 0))

1.12. Keep the notations in 1.10. However we denote the integer N in 1.10 by N_{1}. In 1.12, we let N denotes N_{1} − 1. We assumed Ñ ≥ 3. Then N ≥ 2. Let ω be an automorphism of osp(Ẽ_{0}, , d̄_{N_{1}}) of order 2 defined by:

ω(X)_{ij} = Xî ; ĵ

where

î = \{ \begin{matrix}N′ & if i = N_{1}, \\ N & if i = N′_{1}, \\ i & otherwise. \end{matrix}

Then we can identify osp(Ẽ_{0}, , d̄_{N_{1}})_{0}^{ω} with G(E_{0}, Π_{0}, p_{0}) of (E_{0}, Π_{0}, p_{0}) of the Dynkin diagram:

Diagram 1.12.1.

[figure: B; 1; ε̄_{1} − ε̄_{2}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; ε̄_{N}; .]

Here Hε̄_{j} = d̄_{j}(E_{jj} − E_{j′j′}) (1 ≤ j ≤ N − 1), E_{i} = E_{ii+1}− (−1)^{(p̃(i)p̃(i+1)+p̃(i+1))}g_{i}g_{i+1}E_{(i+1)′i′} (1 ≤ i ≤ N − 2), E_{N_{1}−1} = E_{N_{1}−1N_{1}} + E_{N_{1}−1N′_{1}} −g_{N_{1}−1}g_{N′_{1}}(E_{N_{1}(N_{1}−1)′} + E_{N′_{1}(N_{1}−1)′}), F_{i} = e · \{(−1)^{p̃(i)}E_{i+1i} − (−1)^{p̃(i)p̃(i+1)}g_{i}g_{i+1}E_{i′(i+1)′}\} (1 ≤ i ≤ N_{1} − 2), F_{N_{1}−1} = \frac{1}{2} · e · {(−1)^{p̃(N_{1}−1)}(E_{N_{1}N_{1}−1} + E_{N′_{1}N_{1}−1}) − g_{(N_{1}−1)′}g_{N′_{1}}(E_{(N_{1}−1)′N} + E_{(N_{1}−1)′N′_{1}})}.

Here we introduce the lowest weight ψ of osp(Ẽ_{0}, e)_{1}^{ω}, a lowest weight vector E_{ψ} and a highest weight vector F_{ψ} such that [E_{ψ}, F_{ψ}] = H_{ψ}:

ψ = −ε̄_{1}, F_{ψ} = \frac{1}{2} · e · \{E_{1N_{1}} + E_{1N′_{1}} − g_{1}g_{N_{1}}(E_{N_{1}1} + E_{N′_{1}1})\},

E_{ψ} = E_{N1} + E_{N′1} − (−1)^{p̃(1)}g_{N}g_{1′}(E_{1′N} + E_{1′N′}).

Proposition 1.12.2. osp(Ẽ_{0}, e)_{1}^{ω} is a simple osp(Ẽ_{0}, e)_{0}^{ω}-module.

Proposition 1.12.3. Let (E_{0}, Π_{0}, p_{0}) be the datum of osp(Ẽ_{0}, e) and (E, Π, p) = (E_{0}, Π_{0}, p_{0})^{(ω)}. Put ¯ Ḡ(E, Π, p) = ôsp(Ẽ_{0}, e)^{(ω)}. There is an isomorphism: j : ¯ Ḡ(E, Π, p) → G(E, Π, p) defined by letting j(H ⊗ 1 + ac + bd) = H + aH_{δ} + bH_{Λ_{0}} (H ∈ H) and j(E_{ψ} ⊗ t) = E_{0}, j(F_{ψ} ⊗ t^{−1}) = F_{0}.

Example 1.12.1. If N_{1} ≥ 3, the Dynkin diagrams of (E, Π, p) in Proposition 1.12.1 are:

Diagram 1.12.2. (N ≥ 1)

[figure: (BB); 0; δ − ε̄_{1}; 1; ε̄_{1} − ε̄_{2}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; ε̄_{N}]

[figure: 0; 1]
(∑_{i=1}^{N} p(α_{i}) ≡ 0)

1.13. Let Ẽ_{0} be the C-vector space in 1.5. Here we assume that Ñ = 2N + 2 (N ≥ 1) and shift the numbering of the basis as follows

{ε̄_{0}, ε̄_{1}, …, ε̄_{N}, ε̄_{N+1}, ε̄_{N′}, …, ε̄_{1′}}

where i′ = 2N − i + 2. We also assume:

p̃(0) = 1, p̃(N + 1) = 0, p̃(i) = p̃(i′) (1 ≤ i ≤ N),

and

g_{i}, g_{i′} ∈ {±1} (1 ≤ i ≤ N) g_{i}g_{i′} = (−1)^{p̃(i)}.

In 1.13, we denote this Ẽ_{0} by Ẽ_{01} and we again denote the subspace with the basis {ε̄_{1}, …, ε̄_{N}, ε̄_{N+1}, ε̄_{N′}, …, ε̄_{1′}} by Ẽ_{0}. We denote an element X ∈ sl(Ẽ_{01}, e) by

X = ( \begin{matrix}α & a_{i} \\ b_{i} & X_{ij} \end{matrix}  )

where the sizes of the matrices (α), (a_{i}), (b_{i}) and (X_{ij}) are 1 × 1, 1 × (2N + 1), (2N + 1) × 1 and (2N + 1) × (2N + 1) respectively. Let Ξ be an automorphism of sl(Ẽ_{01}, e) of order 4 defined by:

Ξ(X) = ( \begin{matrix}−α & −\sqrt{−1}g_{i}b_{i′} \\ −\sqrt{−1}g_{i′}a_{i′} & Ω((X_{ij})) \end{matrix} ) .

Then sl(Ẽ_{01}, e)_{0}^{Ξ} consists of the matrices

X = ( \begin{matrix}0 & 0 \\ 0 & X_{ij} \end{matrix}  )

where the 2N + 1 × 2N + 1-matrices (X_{ij}) form osp(Ẽ_{0}, e) whose Dynkin diagram is:

Diagram 1.13.1 [figure: B; 1; ε̄_{1} − ε̄_{2}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; ε̄_{N}; .]

Here we introduce the lowest weight ψ of sl(Ẽ_{01}, e)^{Ξ}, a lowest weight vector E_{ψ} ∈ sl(Ẽ_{01}, e)_{1}^{Ξ} and a highest weight vector F_{ψ} ∈ sl(Ẽ_{01}, e)_{3}^{Ξ} such that [E_{ψ}, F_{ψ}] = H_{ψ}:

ψ = −ε̄_{1}, F_{ψ} = e · {(−1)^{p̃(1)}E_{01′} + g_{1}E_{1′0}},
E_{ψ} = E_{1′0} − g_{1}E_{01′}.

Proposition 1.13.1. sl(Ẽ_{01}, e)_{1}^{Ξ} and sl(Ẽ_{1}, e)_{3}^{Ξ} are the simple osp(Ẽ, e) = sl(Ẽ_{01}, e)_{0}^{Ξ}-modules.

Proposition 1.13.2. As an osp(Ẽ_{0}, e)-module,

sl(Ẽ_{1}, e)_{2}^{Ξ} ≅ sl(Ẽ, e)_{1}^{Ω} ⊕ I

where I = C ∑_{i=1}^{2N+1}{(−1)^{p̃(i)}E_{00} + E_{ii}}.

Proposition 1.13.3. Let (E_{0}, Π_{0}, p_{0}) be the datum of sl(Ẽ_{01}, e) and (E, Π, p) = (E_{0}, Π_{0}, p_{0})^{(Ω)}. Put ¯Ḡ(E, Π, p) = ŝl(Ẽ_{01}, e)^{(Ξ)}. There is an epimorphism: j : ŝl(Ẽ_{0}, e)^{(4)} → G(E, Π, p) defined by letting j(H ⊗ 1 + ac + bd) = H + aH_{δ} + bH_{Λ_{0}} (H ∈ H) and j(E_{ψ} ⊗ t) = E_{0}, j(F_{ψ} ⊗ t^{−1}) = F_{0}.

If ∑_{i=1}^{N} d̄_{i} ≠ 0, then j is an isomorphism. If (−1)Ñ = 1 and ∑_{i=1}^{N} d̄_{i} = 0, then

ker j = ⊕_{i}I ⊗ t^{4i−2}.

.

Example 1.13.1. The Dynkin diagrams of (E, Π, p) in Proposition 1.13.2 are:

Diagram 1.13.2. (N ≥ 1)

[figure: (BB); 0; δ − ε̄_{1}; 1; ε̄_{1} − ε̄_{2}; …; N − 1; ε̄_{N−2} − ε̄_{N−1}; N; ε̄_{N}]

[figure: 0; 1]
(∑_{i=1}^{N} p(α_{i}) ≡ 1)

1.14. We call the datum (E, Π, p) in the following table the one of affine ABCD-type. In the follow-

ing table, a subscript of a name of a Dynkin diagram shows ∑_{i=0}^{n} p(α_{i}) (mod 2) of the corresponding

superalgebra.

Name

Another name  Dynkin diagram  G = ¯ Ḡ
A_{N−1}^{(1)}  ŝl(Ẽ)^{(1)}  (AA)_{0}  ∑_{i=1}^{N} d̄_{i} ≠ 0
B_{N}^{(1)}  ôsp(Ẽ_{odd})^{(1)}  (DB)_{0} (CB)_{1}  all
A_{2N}^{(2)}  ŝl(Ẽ_{odd})^{(2)}  (DB)_{1} (CB)_{0}  all
C_{N}^{(1)} D_{N}^{(1)}  ôsp(Ẽ_{even})^{(1)} (CC)_{0} (CD)_{1} (DD)_{0} (DC)_{1}  all
A_{2N−1}^{(2)}  ŝl(Ẽ_{even})^{(2)} (CC)_{1} (CD)_{0} (DD)_{1} (DC)_{0} ∑_{i=1}^{N} d̄_{i} ≠ 0
D_{N+1}^{(2)}  ôsp(Ẽ_{even})^{(2)}  (BB)_{0}  all
A_{2N+1}^{(4)}  ŝl(Ẽ)^{(4)}  (BB)_{1}  ∑_{i=1}^{N} d̄_{i} ≠ 0

1.15. We call the data whose Dynkin diagrams are Diagram 5.1.4, Diagram 5.2.3 and Diagram 5.3.3 D(2; 1, x)^{(1)}-type, F_{4}^{(1)}-type and G_{3}^{(1)}-type respectively. We call these affine exceptional type.

2. Affine Weyl type isomorphism 2.1. In this section, we introduce a family {L_{i}} of isomorphisms between Lie superalgebras G(E, Π, p) of (E, Π, p) of affine ABCD-type (see also [FSS]). The isomorphism L_{i} can be considered as a super-version of Weyl group action. However our isomorphisms change Dynkin diagrams. We shall also introduce other Lie superalgebras Ḡ(E, Π, p) of (E, Π, p) of affine ABCD-type. We shall show that Ḡ(E, Π, p)’s are universal superalgebras satisfying: (i) G(E, Π, p) is a quotient of Ḡ(E, Π, p), (ii) L_{i} can be lifted to isomorphisms of Ḡ(E, Π, p)’s. Finally we shall show that Ḡ(E, Π, p) = ¯ Ḡ(E, Π, p). We have already introduced ¯ Ḡ(E, Π, p) in a concrete way in §1 for (E, Π, p) of affine ABCD-type. We have known that ¯ Ḡ(E, Π, p) ≠ G(E, Π, p) if and only if G(E, Π, p) =

ŝl(Ẽ)^{(1)}, ŝl(Ẽ_{even})^{(2)}, ŝl(Ẽ)^{(4)} and ∑_{i=1}^{N} d̄_{i} = 0.

Our idea using Weyl-group-type isomorphisms relates to [LS]. 2.2. Let

Ĕ_{ij}(k) = [...[[E_{j}, \underbrace{E_{i}], E_{i}], …, E_{i}]}_{k− times}, F̆_{ij}(k) = [...[[F_{j}, \underbrace{F_{i}], F_{i}], …, F_{i}]}_{k− times} .

The calculations of the following lemma are useful.

Lemma 2.2.1. (i) If i ≠ j, then [[E_{j}, E_{i}], F_{i}] = −(α_{i}, α_{j})E_{j}, [E_{i}, [F_{j}, F_{i}]] = (−1)^{p(α_{i})p(α_{j})}(α_{i}, α_{j})F_{j} [[E_{j}, E_{i}], [F_{j}, F_{i}]] = (−1)^{p(α_{i})p(α_{j})}(α_{i}, α_{j})H_{α_{i}+α_{j}}.

(ii) For the i-th simple root α_{i} ∈ Π satisfying (α_{i}, α_{i}) ≠ 0, let a_{ij} = \frac{2(α_{j}, α_{i})}{(α_{i}, α_{i})}. Assume that a_{ij} is

an even integer if p(α_{i}) = 1. Put

< k; −a_{ij} >= \{ \begin{matrix}k(−a_{ij} − k + 1) & if p(α_{i}) = 0, \\ k & if p(α_{i}) = 1 and k is even, \\ −a_{ij} − k + 1 & if p(α_{i}) = 1 and k is odd. \end{matrix}

We put < k; −a_{ij} >! = ∏_{r=1}^{k} < r; −a_{ij} >. Then

[E_{i}, F̆_{ij}(k)] = −(−1)^{p(α_{i})p(α_{j})} < k; −a_{ij} > F̆_{ij}(k − 1),
[Ĕ_{ij}(k), F_{i}] = (−1)^{(k−1)p(α_{j})} < k; −a_{ij} > Ĕ_{ij}(k − 1),
[Ĕ_{ij}(k), F̆_{ij}(k)] = (−1)^{k}(−1)^{p(α_{i})p(α_{j})} < k; −a_{ij} >! H_{kα_{i}+α_{j}}.

Lemma 2.2.2. Let G = G(E, Π, p) be a Kac-Moody superalgebra with the triangular decomposition G = N^{+} ⊗ H ⊗ N^{−}. If X ∈ N^{+} (resp. Y ∈ N^{−}) satisfies [X, F_{k}] = 0 (resp. [E_{k}, Y] = 0) for any k, then X = 0 (resp. Y = 0) in G. Proof. We can assume that X is in an root space. Let r_{+}(X) be the ideal of N^{+} generated by X. Then r_{+}(X) is an ideal of G such that r_{+}(X) ∩ H = 0. Hence X = 0.

Q.E.D.

As an immediate consequence of Lemma 2.2.1 and Lemma 2.2.2, we have:

Lemma 2.2.3 (i) For the i-th simple root α_{i} ∈ Π satisfying (α_{i}, α_{i}) ≠ 0, let a_{ij} = 2(α_{j}, α_{i})/(α_{i}, α_{i}). Assume that a_{ij} is an even integer if p(α_{i}) = 1. Then Ĕ_{ij}(−a_{ij}) = 0, F̆_{ij}(−a_{ij}) = 0 in G. (ii) If (α_{i}, α_{i}) = 0, then [E_{i}, E_{i}] = [F_{i}, F_{i}] = 0 in G. Proof. Direct calculations.

Proposition 2.2.1. Let G = G(E, Π, p) be a Kac-Moody superalgebra such that an i-th simple root α_{i} ∈ Π satisfies (α_{i}, α_{i}) = 0. Let Π′ = {α′_{1}, … α′_{n}} = {−α_{i}, α_{j} + α_{i} (j ≠ i, (α_{i}, α_{j}) ≠ 0), α_{j} (j ≠ i, (α_{i}, α_{j}) = 0) }. Put G′ = G(E, Π′, p). Then there are isomorphisms ϕ : G → G′ such that

ϕ(H_{γ}) = H_{γ}.  (2.2.1)

(In particular,

ϕ(H_{γ}) = \{ \begin{matrix}−H_{α′_{i}} & if γ = α_{i}, \\ H_{α′_{i}+α′_{j}} & if γ = α_{j} and (α_{i}, α_{j}) ≠ 0, \\ H_{α′_{j}} & if γ = α_{j} and (α_{i}, α_{j}) = 0. \end{matrix}

)

ϕ(E_{i}) = −(−1)^{p(α_{i})}F_{i}, ϕ(F_{i}) = −E_{i},  (2.2.2)

ϕ(E_{j}) = −\frac{(−1)^{p(α_{i})p(α_{j})}}{(α_{i}, α_{j})} [E_{j}, E_{i}], ϕ(F_{j}) = −[F_{j}, F_{i}], ( (α_{i}, α_{j}) ≠ 0 )  (2.2.3)
ϕ(E_{j}) = E_{j}, ϕ(F_{j}) = F_{j} ( i ≠ j, (α_{i}, α_{j}) = 0 )  (2.2.4)

Proof. Let H′ be the Cartan subalgebra of G′. Denote the right hand sides of (2.2.1-4) by H′_{γ}, E′_{j} and F′_{j}. We can show that there is an epimorphism y : G̃(E, Π, p) → G′. such that y(H_{γ}) = H′_{γ}, y(E_{j}) = E′_{j} and y(F_{j}) = F′_{j}. For example, by Lemma 2.2.3 (ii), we can show that the elements satisfy (1.2.1-3). Clearly y_{|H} : H → H′ is isomorphism. Hence there exists an epimorphism ϕ_{1} : G′ → G such that ϕ_{1}(H′_{γ}) = H_{γ}, ϕ_{1}(E′_{j}) = E_{j} and ϕ_{1}(F′_{j}) = F_{j}. Since ϕ_{1}_{|H′} is injective, ϕ_{1} is isomorphism. ϕ_{1} is nothing else but ϕ^{−1}.

Q.E.D.

Keep the notations in the statement of Proposition 2.2.1. We still assume (α_{i}, α_{i}) = 0. For the Dynkin diagram Γ of (E, Π, p), we denote the Dynkin diagram of (E, Π′, p) by Γ^{<i>}. Similarly to the proof of Proposition 2.2.1, we have following Propositions 2.2.2-3.

Proposition 2.2.2. Let G = G(E, Π, p) be a Kac-Moody superalgebra such that an i-th simple root α_{i} ∈ Π satisfies (α_{i}, α_{i}) ≠ 0. Let d_{i} = (α_{i}, α_{i})/2. Let

(−a_{ij})_{s}! = \{ \begin{matrix}(−a_{ij})! & if p(α_{i}) = 0, \\ (\frac{−a_{ij}}{2} )!2^{\frac{−a_{ij}}{2}} & if p(α_{i}) = 1, \end{matrix}

Assume that a_{ij} is an even integer if p(α_{i}) = 1. Then there is an isomorphisms ϕ : G → G such that

ϕ(H_{γ}) = H_{γ− \frac{2(γ,α_{i})}{(α_{i},α_{i})} α_{i}}  (2.2.7)
ϕ(E_{i}) = −(−1)^{p(α_{i})}F_{i}, ϕ(F_{i}) = −E_{j},  (2.2.8)
ϕ(E_{j}) = \frac{1}{(−a_{ij})_{s}!d_{i}^{−a_{ij}}} Ĕ_{ij}(−a_{ij}),  (2.2.9)

ϕ(F_{j}) = (−1)^{−a_{ij}} \frac{1}{(−a_{ij})_{s}!}F̆_{ij}(−a_{ij}).  (2.2.10)

Proposition 2.2.3. For Π = {α_{1}, …, α_{n}}, let a : {1, …, n} → {1, …, n} be a bijective map such that (α_{a(i)}, α_{a(j)}) = (α_{i}, α_{j}) (1 ≤ i, j ≤ n). Then there is an isomorphisms ϕ : G → G such that

H_{α_{i}} = H_{α_{a(i)}},  (2.2.11)
ϕ(E_{i}) = E_{a(i)}, ϕ(F_{i}) = F_{a(i)}.  (2.2.12)

2.3. Here we fix a positive integer N. Let Θ_{N} be the set of affine ABCD-type Dynkin diagrams Γ satisfying that the number of the dots of Γ is N − 1 if Γ is of the affine A-type, N otherwise. Let D(Θ_{N}) be the set of the data (E, Π = {α_{0}, α_{1}, …, α_{n}}, p) whose Dynkin diagrams belong to Θ_{N}. Let Γ ∈ Θ_{N} and (E, Π, p) ∈ D(Θ_{N}). For 0 ≤ i ≤ n, we define Γ^{σ(i)} = Γ′ ∈ Θ_{N} and (E^{σ(i)} = ⊕_{i=1}^{N}Cε̄′_{i} ⊕ Cδ ⊕ CΛ_{0}, Π^{σ(i)}, p^{σ(i)}) ∈ D(Θ_{N}) by following (i)-(iii): (We put d̄′_{i} = (ε̄′_{i}, ε̄′_{i}).)

(i) If 1 ≤ i ≤ N − 1 and (α_{i}, α_{i}) = 0, then (E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) satisfies that Γ^{σ(i)} = Γ^{<i>} (read the sentences before Proposition 2.2.2) and d̄′_{i} = d̄_{i+1}, d̄′_{i+1} = d̄_{i}, d̄′_{j} = d̄_{j} (j ≠ i, i + 1).

(ii) If Γ is affine A type and i = 0, then (E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) satisfies that Γ^{σ(i)} = Γ^{<i>} and d̄′_{1} = d̄_{N}, d̄′_{N} = d̄_{1} and d̄′_{j} = d̄_{j} (j ≠ 1, N).

(iii) Otherwise we put Γ^{σ(i)} = Γ and (E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) = (E, Π, p).

2.4. For (E, Π, p) ∈ D(Θ_{N}), let (E^{†}, Π^{†} = {α_{0}^{†}, …, α_{n}^{†}}, p^{†}) ∈ D(Θ_{N}) be an another datum satisfying:

(i) E^{†} = E,

(ii) For (E, Π, p) ∈ D(Θ_{N}), the type of the Dynkin diagram Γ^{†} of (E^{†}, Π^{†}, p^{†}) ∈ D(Θ_{N}) is:

(AA) if Γ is type (AA),
(BB) if Γ is type (BB),
(CB) if Γ is type (CB) or (DB),
(CC) if Γ is type (CC), (CD), (DC) or (DD).

We shall not need p^{†}. So we merely denote (E^{†}, Π^{†}, p^{†}) by (E^{†}, Π^{†}).

Remark 2.4.1. We can easily see that two Γ^{†}’s defined for (E, Π, p) and (E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) are same.

Lemma 2.4.1. P_{+} = Z_{+}α_{0} ⊕ … ⊕ Z_{+}α_{n} ⊂ P_{+}^{†} = Z_{+}α_{0}^{†} ⊕ … ⊕ Z_{+}α_{n}^{†}.

2.5. On E = ⊕_{i=1}^{N}Cε̄_{i} ⊕ Cδ ⊕ CΛ_{0}, we define another symmetric form (( , )) : E × E → C by

((ε̄_{i}, ε̄_{i})) = δ_{ij}, ((ε̄_{i}, δ)) = 0, ((δ, δ)) = 0, ((δ, Λ_{0})) = 1, ((Λ_{0}, Λ_{0})) = 1.

For (E, Π, p) ∈ D(Θ_{N}) and 0 ≤ i ≤ n, define I^{σ(i)} : E = ⊕_{i=1}^{N}Cε̄_{i} ⊕ Cδ ⊕ CΛ_{0} → E^{σ(i)} = ⊕_{i=1}^{N}Cε̄′_{i} ⊕ Cδ ⊕ CΛ_{0} by I^{σ(i)}(ε̄_{i}) = ε̄′_{i}, I^{σ(i)}(δ) = δ and I^{σ(i)}(Λ_{0}) = Λ_{0}. We also define a linear map σ(i) : E → E^{σ(i)} by

σ(i) (v) = I^{σ(i)}(v − \frac{2((v, α_{i}^{†}))}{((α_{i}^{†}, α_{i}^{†}))}α_{i}^{†}).

As an easy consequence from Propositions 2.2.1-3, we have:

Theorem 2.5.1. For (E, Π, p) ∈ D(Θ_{N}) and 0 ≤ i ≤ n, Let H ⊕ ⊕_{α∈Φ^{σ(i)}}G_{α}^{σ(i)} be the root space decomposition of G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}).

(i) There is an isomorphism L_{i} : G(E, Π, p) → G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) such that:

L_{i}(H_{γ}) = H_{σ(i)(γ)} (γ ∈ E).  (2.5.1)

In particular, L_{i} satisfies:

L_{i}(G_{α}) = G_{σ(i)(α)}^{σ(i)} (α ∈ Φ).  (2.5.2)

(ii) Let L′_{i} and L_{i} be two isomorphisms satisfying (2.5.1) Then there exist a_{j} ∈ C^{∗} such that

L′_{i}(E_{j}) = a_{j}L_{i}(E_{j}), L′_{i}(F_{j}) = a_{j}^{−1}L_{i}(F_{j}).  (2.5.3)

Proof. (i) We can choose one of ϕ’s in Propositions 2.2.1-3 as L_{i}. It is obvious L_{i} satisfies (2.5.1-2).

(ii) By (2.5.2) and the fact of dim G_{α_{j}} = 1, dim G_{σ(i)(α_{j})}^{σ(i)} = 1. By the fact of [L_{i}(E_{j}), L_{i}(F_{j})] = H_{σ(i)(α_{j})} = [L′_{i}(E_{j}), L′_{i}(F_{j})], we get (2.5.3).

Remark 2.5.1. For example, σ(i) and L_{i} move as follows:

[figure: i − 1; i; i + 1; i; i − 1; i; i + 1; ,; i − 1; i; i + 1; i; i − 1; i; i + 1; (1 ≤ i ≤ N − 1),]

[figure: N − 2; N − 1; N; N − 1; N − 2; N − 1; N; ,]

[figure: N − 2; N − 1; N; N; N-2; N − 2; N − 1; N; N; N-1; N-1; N − 2; N − 1; N]

Proposition 2.5.1. For (E, Π, p) ∈ D(Θ_{N}), let G = G(E, Π, p). Then dim G_{α} = 1 if α ∈ Φ\Zδ. Proof. We consider the case of α ∈ Φ_{+} \ Z_{+}δ. We use an induction on the height with respect to Π^{†}. By ((α, α)) > 0, ((α, α_{i})) > 0 for some i. If α ∉ Zα_{i} then the height of σ(i)(α) is smaller than the height of α. Thus by σ(i)’s, we can get a path from α to rα_{i} ∈ Π′ of another datum (E′, Π′, p′) where r ∈ {1, 2} if p(α_{i}) = 1 and (α_{i}, α_{i}) ≠ 0, r = 1 otherwise. On the other hand, dim G′_{rα_{j}} = 1 by Lemma 1.2.1.

Q.E.D.

2.6. Let D(Θ_{N})_{(XY)} denote {(E, Π, p) ∈ Θ_{N}|(E, Π, p) is (XY) − type}. Then {σ(i)|1 ≤ i ≤ n} preserve D(Θ_{N})_{1} = D(Θ_{N})_{(AA)}, D(Θ_{N})_{2} = D(Θ_{N})_{(BB)}, D(Θ_{N})_{3} = D(Θ_{N})_{(CB)} ∪ D(Θ_{N})_{(DB)} or D(Θ_{N})_{4} = D(Θ_{N})_{(CC)} ∪D(Θ_{N})_{(CD)} ∪D(Θ_{N})_{(DC)} ∪D(Θ_{N})_{(DD)}. Let W = W (i) denote a group generated by {σ(i)|1 ≤ i ≤ n} acting on D(Θ_{N})_{i}. Then W (i) is isomorphic to the affine Weyl group of the corresponding (E^{†}, Π^{†}).

Let D(Θ_{N})_{i} = ∪_{j=1}^{a_{i}}D(Θ_{N})_{ij} be the orbit decomposition. If a datum (E, Π, p) associated to a Dynkin diagram Γ ∈ Θ_{N} belongs to D(Θ_{N})_{ij}, then we denote D(Θ_{N})_{ij} by D(Θ_{N})[Γ].

2.7. Proposition 2.7.1. Fix an orbit D(Θ_{N})[Γ]. Fix an isomorphism Ł_{i} : G(E, Π, p) → G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) for each (E, Π, p) and σ(i). Then there exists a unique family {Ḡ(E, Π, p) |(E, Π, p) ∈ D(Θ_{N})[Γ]} of Lie superalgebras satisfying following (1), (2) and (3).

(1) For (E, Π, p) ∈ D(Θ_{N})[Γ], there is a sequence of epimorphisms

G̃(E, Π, p) ⟶^{Ψ̃(E,Π,p)} Ḡ(E, Π, p) ⟶^{Ψ(E,Π,p)} G(E, Π, p)

( H_{γ}, E_{i}, F_{i} ⟶ H_{γ}, E_{i}, F_{i} ⟶ H_{γ}, E_{i}, F_{i}).

(2) For the isomorphism L_{i} : G(E, Π, p) → G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}), there is an isomorphism L̄_{i} : Ḡ(E, Π, p) → Ḡ(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) satisfying the following commuting diagram:

[figure: ^{Ḡ(E, Π, p)}; L̄_{i}; ^{Ḡ(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})}; Ψ(E,Π,p); Ψ(E^{σ(i)},Π^{σ(i)},p^{σ(i)}); G(E, Π, p); L_{i}; G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})]

(3) If there is a family {Ḡ^{(λ)}(E, Π, p) |(E, Π, p) ∈ D(Θ_{N})[Γ]} of Lie superalgebras satisfying (1) and (2), then there are epimorphisms

Ψ̄^{(λ)}(E, Π, p) : Ḡ(E, Π, p) → → G̃^{(λ)}(E, Π, p) ((E, Π, p) ∈ D(Θ_{N})[Γ])

satisfying following commutative diagram

Diagram 2.7.1.

[figure: ^{Ḡ(E, Π, p)}; L̄_{i}; ^{Ḡ(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})}; Ψ(E,Π,p); Ψ(E^{σ(i)},Π^{σ(i)},p^{σ(i)}); G(E, Π, p); L_{i}; G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}); ^{Ψ̄^{(λ)}^{(E,Π,p)}}; ^{Ḡ^{(λ)}(E, Π, p)}; Ψ^{(λ)}(E,Π,p); L̄_{i}^{(λ)}; ^{Ψ̄^{(λ)}^{(E^{σ(i)},Π^{σ(i)},p^{σ(i)})}}; ^{Ḡ^{(λ)}(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})}; Ψ^{(λ)}(E^{σ(i)},Π^{σ(i)},p^{σ(i)})]

(ii) The set {Ḡ(E, Π, p) |(E, Π, p) ∈ D(Θ_{N})[Γ]} does not depend on the choice of {L_{i}}.

Proof. (i) Let { C^{(λ)} = {Ḡ^{(λ)}(E, Π, p)|(E, Π, p) ∈ D(Θ_{N})[Γ]} }_{(λ)∈(Λ)} be the family of the families C^{(λ)} ((λ) ∈ (Λ)) satisfying (1) and (2). Let Ψ̃^{(λ)}(E, Π, p) : G̃(E, Π, p) → Ḡ^{(λ)}(E, Π, p) be the epimorphism in (1) for (λ) ∈ (Λ). Let r̄^{(λ)}(E, Π, p) = ker Ψ̃^{(λ)}(E, Π, p) and r̄(E, Π, p) = ∩_{(λ)∈(Λ)}r̄^{(λ)}(E, Π, p). Put Ḡ(E, Π, p) = G̃(E, Π, p)/r̄(E, Π, p). Let [L_{i}(E_{j})], [L_{i}(F_{j})] ∈ Ḡ(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})_{σ(i)(α_{j})} be representatives of L_{i}(E_{j}), L_{i}(F_{j}). Similarly to Proposition 2.5.1, we can show dim Ḡ^{(λ)}(E, Π, p)_{α} = 1 if α ∈ Φ \ Zδ. Then r̄^{(λ)}(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})_{σ(i)(α_{j})}

= r(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})_{σ(i)(α_{j})}. Hence [L_{i}(E_{j})] ≡ L̄_{i}^{(λ)}(E_{j}), [L_{i}(F_{j})] ≡ L̄_{i}^{(λ)}(F_{j}) (mod r̄^{(λ)}(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})). Hence, in Ḡ^{(λ)}(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}), the elements H_{σ(i)(γ)}, [L_{i}(E_{j})] and [L_{i}(F_{j})] satisfy (1.2.1-3) for (E, Π, p) whence, even in Ḡ(E, Π, p), the elements satisfy (1.2.1-3). Hence there is a morphism ˜L̄_{i} : G̃(E, Π, p) → Ḡ(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) such that

Ψ̄^{(λ)}(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) ∘ ˜ L̄_{i} = L̄_{i}^{(λ)} ∘ Ψ̃^{(λ)}(E, Π, p).  (2.7.1)

Therefore there exists L̄_{i} : Ḡ(E, Π, p) → Ḡ(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) such that ˜ L̄_{i} = L̄_{i} ∘ Ψ̃(E, Π, p). Clearly L̄_{i} satisfying the commutative diagram of (2). Using Lemma 2.2.1, we can see that L̄_{i} ∘ L̄_{i} : Ḡ(E, Π, p) → Ḡ(E, Π, p) satisfies that L̄_{i} ∘ L̄_{i}(H_{γ}) = (H_{γ}), and L̄_{i} ∘ L̄_{i}(E_{j}), L̄_{i} ∘ L̄_{i}(F_{j}) are nonzero scalar multiples of E_{j}, F_{j} respectively. Hence L̄_{i} is an isomorphism. (ii) By theorem 2.5.1, if L′_{i} is another L_{i}, then L′_{i}(E_{j}) = a_{j}L_{i}(E_{j}), L′_{i}(F_{j}) = a_{j}^{−1}L_{i}(F_{j}) for some a_{j} ∈ C \ {0} (0 ≤ j ≤ n). Let ϕ_{a} : G(E, Π, p) → G(E, Π, p) be an isomorphism defined by ϕ_{a}(E_{j}) = a_{j}E_{j}, ϕ_{a}(F_{j}) = a_{j}^{−1}F_{j}, ϕ_{a}(H) = H. Then L′_{i} = L_{i} ∘ ϕ_{a}. The ideal r̄(E, Π, p) in the proof of (i) satisfies H ∩ r̄(E, Π, p) = 0. Then r̄(E, Π, p) is the homogeneous ideal. Then we can also define an isomorphism ϕ̄_{a} : Ḡ(E, Π, p) → Ḡ(E, Π, p) similar to ϕ_{a}. Denote L̄_{i} defined for L′_{i} by L̄′_{i}. By the universality of L̄_{i}, it follows that L̄′_{i} = L̄_{i} ∘ ϕ̄_{a}. In particular, Ḡ(E, Π, p) is determined independently of the choice of L_{i}.

Q.E.D.

Lemma 2.7.1. Let (E, Π, p) ∈ D(Θ_{N}) and Ḡ = Ḡ(E, Π, p). Let N̄^{+}, N̄^{−} be the subalgebras of Ḡ generated by E_{i}, F_{i} respectively. Then we have the triangular decomposition

Ḡ = N̄^{+} ⊕ H ⊕ N̄^{−}.

Here H can be identified with H of G. We have N̄^{+} ≅ N̄^{−} (E_{i} ↔ F_{i}). Proof. The triangular decomposition is clear because r̄ ∩ H = 0. Let r̄ be the ideal r̄(E, Π, p) of G̃(E, Π, p) in the proof of Proposition 2.7.1. Let r̄_{±} = r̄ ∩ Ñ^{±}. Let r̄_{−}^{1} (resp. r̄_{+}^{1}) be the ideal defined as the image of r̄_{+} of the map Ñ^{+} → Ñ^{−} (E_{i} → F_{i}) (resp. Ñ^{−} → Ñ^{+} (F_{i} → E_{i}) ). Put r̄^{1} = r̄_{−}^{1} ⊕ r̄_{−}^{1} and Ḡ^{1} = G̃/r̄^{1}. By the universality, we can show Ḡ^{1} = Ḡ. Then we have r̄_{∓}^{1} = r̄_{±}

Q.E.D.

Lemma 2.7.2. For α_{i} ∈ Π, we have dim Ḡ_{α_{i}} = 1. If p(α_{i}) = 1 and (α_{i}, α_{i}) ≠ 0, then dim G_{2α_{i}} = 1. Proof. The proof is obtained similarly to the proof of Lemma 1.2.1.

Q.E.D.

Proposition 2.7.2. Let Φ be the set of the roots of G(E, Π, p) associated with (E, Π, p) ∈ D(Θ_{N}). For Ḡ = Ḡ(E, Π, p), let Ḡ_{γ} = {x ∈ Ḡ | [H, x] = γ(H)x}. Then we have

(i) dim Ḡ_{γ} = 1 if γ ∈ Φ \ Zδ. (ii) dim Ḡ_{γ} ≥ dim G_{γ} if γ ∈ Zδ. (iii) dim Ḡ_{γ} = 0 if γ ∉ Φ ∪ {0}.

In particular,

ker Ψ(E, Π, p) ⊂ ⊕_{r≠0}Ḡ_{rδ}.

Proof. By Lemma 2.7.1, we may assume γ ∈ P_{+}. (ii) is clear. Using Lemma 2.7.2, we can proof (i) similarly to the proof of Proposition 2.5.1. Moreover (iii) can be proved similarly to the proof of Proposition 2.5.1: If γ ∉ Φ ∪ {0}, then, by σ(i)’s, we can get a path from γ to (⊕_{α_{i}∈Π′}Zα_{i}) \ (P′_{+} ∪ −P′_{+}) of another datum (E′, Π′, p′). By the triangular decomposition of Ḡ(E′, Π′, p′), we have dim Ḡ_{γ} = 0.

Q.E.D.

2.8. Proposition 2.8.1. Let (E, Π, p) ∈ D(Θ_{N}).

(i) There exists a unique Lie superalgebra Ḡ^{‡}(E, Π, p) satisfying following (1), (2) and (3).

(1) For (E, Π, p) ∈ D(Θ_{N}), there is a sequence of epimorphisms

G̃(E, Π, p) ⟶^{Ψ̃^{‡}(E,Π,p)} Ḡ^{‡}(E, Π, p) ⟶^{Ψ^{‡}(E,Π,p)} G(E, Π, p)

( H_{γ}, E_{i}, F_{i} ⟶ H_{γ}, E_{i}, F_{i} ⟶ H_{γ}, E_{i}, F_{i}).

(2) By (1), there is a root space decomposition

Ḡ^{‡}(E, Π, p) = H ⊕ (⊕_{α∈Ψ^{‡}} Ḡ_{α}^{‡})

such that E = H^{∗} ⊃ Ψ^{‡} ⊃ Ψ. Here Ψ is the set of the roots of G(E, Π, p). Then the assumption (2) is that Ψ^{‡} = Ψ and dim Ḡ_{α}^{‡} = 1 if α ∈ Ψ \ Zδ.

(3) If there is a Lie superalgebra Ḡ^{‡(λ)}(E, Π, p) satisfying (1) and (2), then there is an epimorphism:

Ψ̄^{‡(λ)}(E, Π, p) : Ḡ^{‡}(E, Π, p) → → Ḡ^{‡(λ)}(E, Π, p)

(ii) Ḡ^{‡}(E, Π, p) is isomorphic to Ḡ(E, Π, p).

Proof. (i) Let {Ḡ^{‡(λ)}(E, Π, p)}_{(λ)∈(Λ)} be the family of Lie superalgebras satisfying (1) and (2). Let Ψ̃^{‡(λ)}(E, Π, p) : G̃(E, Π, p) → Ḡ^{‡(λ)}(E, Π, p) be the epimorphism in (1) for (λ) ∈ (Λ). Let r̄^{‡(λ)}(E, Π, p) = ker Ψ̃^{‡(λ)}(E, Π, p) and r̄^{‡}(E, Π, p) = ∩_{(λ)∈(Λ)}r̄^{‡(λ)}(E, Π, p). Put Ḡ^{‡}(E, Π, p) = G̃(E, Π, p)/r̄^{‡}(E, Π, p). Then Ḡ^{‡}(E, Π, p) satisfies (1), (2) and (3).

(ii) By the universality of Ḡ^{‡}(E, Π, p), there is an epimorphism L_{i}^{‡} : Ḡ^{‡}(E, Π, p) → Ḡ^{‡}(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}) such that L_{i}^{‡}(H_{γ}) = H_{σ(i)(γ)}, L_{i}^{‡}(E_{j}) = Ψ^{‡}(E, Π, p)^{−1}L_{i}(E_{j}). L_{i}^{‡}(E_{j}) = Ψ^{‡}(E, Π, p)^{−1}L_{i}(E_{j}). Clearly {L_{i}} satisfy (2) of Proposition 2.7.1. Then we have an epimorphism Ḡ(E, Π, p) → Ḡ^{‡}(E, Π, p). By the universality of Ḡ^{‡}(E, Π, p), this map is isomorphism.

Q.E.D.

As an immediate consequence, we have: Lemma 2.8.1. For (E, Π, p) ∈ D(Θ_{N}), there is an epimorphism

Ψ^{†}(E, Π, p) : Ḡ(E, Π, p) → → ¯ Ḡ(E, Π, p).

where ¯ Ḡ(E, Π, p) has already introduced in §1 for (E, Π, p) of affine ABCD-type.

In §3, we will show that Ψ^{†}(E, Π, p) is isomorphism.

3. The estimation of dim Ḡ_{rδ} 3.1. Proposition 3.1.1. (i) Let (E, Π, p) ∈ D(Θ_{N}). Put D_{N} = ∑_{i=1}^{N} d̄_{i}. Then

(δ, 2ρ) = \{ \begin{matrix}2D_{N} & (AA), \\ 2D_{N} & (BB), \\ 2d̄_{1} + 4D_{N} & (CB), \\ −2d̄_{1} + 4D_{N} & (DB), \\ 2d̄_{1} + 4D_{N} + 2d̄_{N} & (CC), \\ −2d̄_{1} + 4D_{N} + 2d̄_{N} & (DC), \\ 2d̄_{1} + 4D_{N} − 2d̄_{N} & (CD), \\ −2d̄_{1} + 4D_{N} − 2d̄_{N} & (DD). \end{matrix}

(ii) For exceptional type, we have:

(δ, 2ρ) = \{ \begin{matrix}0 & D(2, 1,; x)^{(1)}, \\ −12 & F_{4}^{(1)}, \\ 12 & G_{3}^{(1)}. \end{matrix}

Proof. Direct calculations. Q.E.D.

Here we again remark, if there is a relation of weight β ∈ P_{+} such that (β, β) ≠ 2(β, ρ), then the relation can be obtained by relations of lower weights (see Proposition 1.2.1). Therefore have: Lemma 3.1.1 If (E, Π, p) ∈ D(Θ_{N}) satisfies (δ, δ) ≠ 2(δ, ρ), then Ḡ(E, Π, p) ≅ G(E, Π, p). 3.2. Proposition 3.2.1. Let (E, Π, p) ∈ D(Θ_{N})[Γ]. Assume that x-th simple root α_{x} ∈ Π and y-th simple root α_{y} ∈ Π satisfy α_{x} + α_{y} ∈ Φ_{+} (i.e. (α_{x}, α_{y}) ≠ 0), α_{x} + α_{y} = σ(x)(α_{y}^{†}) and that Π_{(x,y)} = (Π \ {α_{x}, α_{y}}) ∪{α_{x} + α_{y}} is affine ABCD type.

(i) Let Π_{(x,y)}^{†} be Π^{†} of Π_{(x,y)}. Then Π_{(x,y)}^{†} = (Π^{†} \ {α_{x}^{†}, α_{y}^{†}}) ∪σ(x)(α_{y}^{†}) Let W_{(x,y)} denote a subgroup of W generated by ({σ(0), …, σ(n)} \ {σ(x), σ(y)}) ∪{σ(x)σ(y)σ(x)}. Then W_{(x,y)} is W defined for (E, Π_{(x,y)}, p).

(ii) Let (E, Π_{(x,y)}, p) be a datum such that the set of simple roots is Π_{(x,y)}. Then there is a homomorphism i : Ḡ(E, Π_{(x,y)}, p) → Ḡ(E, Π, p) such that

i(H_{α_{j}}) = \{ \begin{matrix}H_{α_{x}+α_{y}} & α_{j} = α_{x} + α_{y}, \\ H_{α_{j}} & α_{j} ≠ α_{x} + α_{y}, \end{matrix}

i(E_{j}) = \{ \begin{matrix}(−1)^{p(α_{x})p(α_{y})}(α_{x}, α_{y})^{−1}[E_{x}, E_{y}] & α_{j} = α_{x} + α_{y}, \\ E_{j} & α_{j} ≠ α_{x} + α_{y}, \end{matrix}

i(F_{j}) = \{ \begin{matrix}[F_{x}, F_{y}] & α_{j} = α_{x} + α_{y}, \\ F_{j} & α_{j} ≠ α_{x} + α_{y}. \end{matrix}

Proof. (i) We can check the fact for each affine type.

(ii) Let {L̄_{0}, …, L̄_{n}} be the isomorphisms defined in Proposition 2.7.1. We consider an orbit Orbit(Ḡ(E, Π, p))_{(x,y)} through Ḡ(E, Π, p) under the action of a subgroup generated by ({L̄_{0}, …, L̄_{n}} \ {L̄_{x}, L̄_{y}}) ∪ {L̄_{x}L̄_{y}L̄_{x}}. Then Orbit(Ḡ(E, Π, p))_{(x,y)} satisfies the conditions of {Ḡ^{(λ)}(E, Π_{(x,y)}, p)} in Proposition 2.7.1 where {L̄_{i}^{(λ)}} is ({L̄_{0}, …, L̄_{n}} \ {L̄_{x}, L̄_{y}}) ∪ {L̄_{x}L̄_{y}L̄_{x}}. By the universality of Ḡ(E, Π_{(x,y)}, p), we get the homomorphism i : Ḡ(E, Π_{(x,y)}, p) → Ḡ(E, Π, p).

Q.E.D.

3.3. Let (E, Π, p) ∈ D(Θ_{N})[Γ]. Here we shall show that Ḡ(E, Π, p) is isomorphic to ¯ Ḡ(E, Π, p). By Proposition 2.8.1, we have to show dim ¯ Ḡ(E, Π, p)_{nδ} = dim Ḡ(E, Π, p)_{nδ}. Here we only prove the fact in the case of (E, Π, p) of Diagram 1.11.6 with ∑ p(α_{i}) ≡ 1. Because we can prove the fact in another case by the similar way. In this case, ¯ Ḡ = ôsp(Ẽ_{0}, e)^{(I)} (see 1.11). Then we have to show:

dim Ḡ(E, Π, p)_{nδ} ≤ N (n ≠ 0).  (3.3.1)

We start with N = 3. In this case, its Dynkin diagram is:

Diagram 3.3.1

[figure: 0; 1; 2; 3]

The isomorphism σ(1) divert Diagram 3.3.1 into:

Diagram 3.3.2

[figure: 0; 2; 1; 3]

Therefore

it is sufficient to show (3.3.1) for (E, Π, p) of Diagram 3.3.2. By Proposition 3.2.1, we have the homomorphism

i : Ḡ(E, Π_{(2,3)}, p) → Ḡ(E, Π, p).

Here the Dynkin diagram of (E, Π_{(2,3)}, p) is:

Diagram 3.3.3

[figure: α_{0}; α_{1}; α_{2} + α_{3}]

This is equivalent to Diagram 1.6.2 as the Dynkin diagram of the Kac-Moody Lie superalgebra. By Lemma 3.1.1, Ḡ(E, Π_{(2,3)}, p) = G(E, Π_{(2,3)}, p). Hence

dim Ḡ(E, Π_{(2,3)}, p)_{nδ} = 2 (n ≠ 0).  (3.3.2)

We assume n > 0. For a root γ ∉ Zδ of G(E, Π, p) (resp. G(E, Π_{(2,3)}, p)), let E_{γ} (resp. E_{γ}^{(2,3)}) denote a non-zero element of Ḡ(E, Π, p)_{γ} (resp. Ḡ(E, Π_{(2,3)}, p)_{γ}). By (3.3.2),

{[E_{nδ−α_{0}}^{(2,3)}, E_{α_{0}}^{(2,3)}], [E_{nδ−α_{1}}^{(2,3)}, E_{α_{1}}^{(2,3)}], [E_{nδ−(α_{2}+α_{3})}^{(2,3)}, E_{α_{2}+α_{3}}^{(2,3)}]} are linearly dependent. Transposing by i, we see that {[E_{nδ−α_{0}}, E_{0}], [E_{nδ−α_{1}}, E_{1}], [E_{nδ−(α_{2}+α_{3})}, [E_{2}, E_{3}]]} are linearly dependent. Since [E_{nδ−(α_{2}+α_{3})}, [E_{2}, E_{3}]] = [[E_{nδ−(α_{2}+α_{3})}, E_{2}], E_{3}] −(−1)^{p(α_{2})p(α_{3})}[[E_{nδ−(α_{2}+α_{3})}, E_{3}], E_{2}], {[E_{nδ−α_{0}}, E_{0}], [E_{nδ−α_{1}}, E_{1}], [E_{nδ−α_{2}}, E_{2}], [E_{nδ−α_{3}}, E_{3}]} are linearly dependent. Hence dim Ḡ(E, Π, p) ≤ 3. Then we could show (3.3.1) for N = 3.

Next we show (3.3.1) for N ≥ 4 by induction. By Proposition 3.2.1, we can use the homomorphism

i : Ḡ(E, Π_{(N−3,N−2)}, p) → Ḡ(E, Π, p).

Using a similar argument to that in the case of N = 3, we can show (3.3.1).

Using a similar argument to that in the above case, we can get:

Theorem 3.3.1. Let (E, Π, p) ∈ D(Θ_{N}). Then Ḡ(E, Π, p) is isomorphic to ¯ Ḡ(E, Π, p).

4. Relations of Affine ABCD-types 4.1. Let (E, Π, p) ∈ D(Θ_{N}). Using the definition of Ḡ(E, Π, p) given in Proposition 2.7.1, we can directly calculate defining relations of Ḡ(E, Π, p).

Theorem 4.1.1. Let (E, Π, p) ∈ D(Θ_{N}) (i.e., (E, Π, p) is affine ABCD type). The Lie superalgebra Ḡ(E, Π, p) is defined by generators H ∈ H, E_{i}, F_{i} (0 ≤ i ≤ n) with parities p(H) = 0, p(E_{i}) = p(E_{i}) = p(α_{i}) and relations:

(1) [H, H′] = 0, (H, H′ ∈ H)

(2) [H, E_{i}] = α_{i}(H)E_{i}, [H, F_{i}] = −α_{i}(H)F_{i},

(3) [E_{i}, F_{j}] = δ_{ij}H_{α_{i}},

(4) Relations of E_{i}’s.

(i) [E_{i}, E_{j}] = 0

if (α_{i}, α_{j}) = 0, (i ≠ j),

(ii) [E_{i}, E_{i}] = 0

[figure: if; i]

(iii) [E_{i}, [E_{i}, …, [E_{i}, E_{j}]…]] = 0 (E_{i} appears 1 − 2(α_{i}, α_{j})/(α_{i}, α_{i}) times)

if (α_{i}, α_{i}) ≠ 0 and (−1)^{\{p(α_{i}) \frac{2(α_{j} ,α_{i})}{(α_{i},α_{i})} \}} = 1,

(iv) [[[E_{i}, E_{j}], E_{k}], E_{j}] = 0

[figure: if; i; −x; j; x; k; (x ≠ 0),]

(v) [[[E_{i}, E_{j}], [[E_{i}, E_{j}], E_{k}]], E_{j}] = 0

[figure: if; i; j; k; ,]

(vi) [[[[[[E_{i}, E_{j}], E_{k}], E_{l}], E_{k}], E_{j}], E_{k}] = 0

[figure: if; i; j; k; l; ,]

(vii) (−1)^{p(α_{i})p(α_{k})}(α_{i}, α_{k})[[E_{i}, E_{j}], E_{k}] = (−1)^{p(α_{i})p(α_{j})}(α_{i}, α_{j})[[E_{i}, E_{k}], E_{j}]

[figure: if; .; i; a; b; .; j; c; .; k]
(a = (α_{i}, α_{j}), b = (α_{i}, α_{k}), c = (α_{j}, α_{k}),
abc ≠ 0, a + b + c = 0,
p(α_{i})p(α_{j}) + p(α_{i})p(α_{j}) + p(α_{i})p(α_{j}) ≡ 1),

(viii) [[[E_{i}, E_{j}], [E_{j}, E_{k}]], [E_{j}, E_{l}]] = [[[E_{i}, E_{j}], [E_{j}, E_{l}]], [E_{j}, E_{k}]]

[figure: if; i; j; ∗; k; l]

(ix)

\begin{matrix}[[[E_{k}, [E_{l}, [E_{k}, E_{j},]]], [E_{k}, [E_{l}, [E_{k}, [E_{j}, E_{i},]]]]], E_{j}] \\ = 2[[E_{k}, E_{j}], [[E_{k}, [E_{j}, E_{i}]], [E_{k}, [E_{l}, [E_{k}, [E_{j}, E_{i},]]]]]] \end{matrix}

[figure: if; i; j; k; l; ,]

(x)

[E_{j}, [E_{k}, [E_{j}, [E_{k}, E_{i}]]]] = [E_{k}, [E_{j}, [E_{k}, [E_{j}, E_{i}]]]]

[figure: if; k; j ,; i]

(5) Relations of F_{i}’s defined as the same relations as (4). Proof. Direct calculations.

Q.E.D.

5. Relations of D(2; 1, x)^{(1)}, F_{4}^{(1)} and G_{3}^{(1)}

5.1. Let (E_{0}, Π_{0}, p_{0}) = (E_{0} = ⊕_{i=1}^{3}Cα_{i}, Π_{0} = {α_{1}, α_{2}, α_{3}}, p_{0}) be the datum whose Dynkin diagram is:

Diagram 5.1.1.

[figure: 1; −1; 2; −x; 3] .

Let (E, Π, p) = (E_{0}, Π_{0}, p_{0})^{(I)} (see 1.5). Then its Dynkin diagram is:

Diagram 5.1.2.

[figure: 0; x + 1; 1; −1; 2; −x; 3] .

Using the same argument as the one for affine ABCD-type, we can calculate the defining relation of G(E, Π, p). In the argument, (E^{†}, Π^{†}) = (E^{†} = ⊕_{i=0}^{3}Cα_{i}^{†}, Π^{†} = {α_{i}^{†}, (0 ≤ i ≤ 3)}) is defined by a Dynkin diagram:

Diagram 5.1.3.

[figure: 0; 1; 2; 3; ,]

Then the Weyl-group-type isomorphism σ(i) and L_{i} (0 ≤ i ≤ 3) move as follows; Diagram 5.1.4. [figure: 0; x + 1; 1; −1; 2; −x; 3; 0; x + 1; 1; −x; 2; −1; 3; 0; −(x + 1); x; 1; 1; 2; −(x + 1); x; 1; 3; 0; −(x + 1); 1; x; x; 2; −(x + 1); 1; 1; 3; 2; −1; 0; −x; 1; 3; x + 1; 2; −x; 0; −1; 1; 3; x + 1; 1; 1; 2; 2; 0; 3; 0; 3; .]

Theorem 5.1.1. Let (E, Π, p) be the data of which Dynkin diagrams in Diagram 5.1.4. Then Ḡ(E, Π, p) = G(E, Π, p) and its defining relations are ones defined by replacing (vii) of Theorem 4.1.1 with:

(vii) [[[E_{i}, E_{j}], [E_{j}, E_{k}]], [E_{j}, E_{l}]] = x[[[E_{i}, E_{j}], [E_{j}, E_{l}]], [E_{j}, E_{k}]]

[figure: if; i; x + 1; j; −1; −x; k; l; (x ≠ ±1, 0).]

5.2. The same argument can still apply to affine F -type. Let (E_{0}, Π_{0}, p_{0}) = (E_{0} = ⊕_{i=1}^{4}Cα_{i}, Π_{0} = {α_{1}, α_{2}, α_{3}, α_{4}}, p_{0}) be the datum whose Dynkin diagram is:

Diagram 5.2.1.

[figure: 1; 2; 3; 4; ⟹; ×] .

Using the same argument as the one for affine ABCD-type, we can calculate the defining relation of G(E, Π, p). In the argument, (E^{†}, Π^{†}) = (E^{†} = ⊕_{i=0}^{3}, Cα_{i}^{†}, Π^{†} = {α_{i}^{†}, (0 ≤ i ≤ 4)}) is defined by a Dynkin diagram:

Diagram 5.2.2.

[figure: 0; 1; 2; 3; 4; ⟹] .

Then the affine F_{4} Weyl group type isomorphism σ(i) and L_{i} (0 ≤ i ≤ 4) move as follows: (In the diagrams, i + j + ⋯ denote α_{i}^{†} + α_{j}^{†} + · · ·.)

Diagram 5.2.3.

[figure: 1; 2; 3; 4; 0+1+2+3; ⟹; ×; 4; 0; 1; 2; 3; 4; 0+1+2+3; ×; ×; ×; 3; 0; ∗; ∗; 1; 2+3; 3; 0+1+2; 4; ×; ×; 0; 2; 0+1; 2; 3; 4; 1+2+3; ⟹; ×; 4; 1; 0+1; 2; 3; 4; 1+2+3; ×; ×; ×; 3; 1; ∗; ∗; 1+2; 3; 2+3; 0+1; 4; ×; ×; 1; 0; 1+2; 3; 4; 2+3; ⟹; ×; 4; 2; 0; 1+2; 3; 4; 2+3; ×; ×; ×; 3; 2; ∗; ∗; 2; 3; 1+2+3; 0; 4; ×; ×; 0; 1; 2+3; 4; 3; ⟹; ×; 4; 0; 1; 2+3; 4; 3; ×; ×; ×; 3; 0; 1; 2; 3; 4; ×]

Theorem 5.2.1 Let (E, Π, p) be the data of which Dynkin diagrams existing in Diagram 5.2.3. Then Ḡ(E, Π, p) = G(E, Π, p) and its relations are defined by adding the following relations to the ones of Theorem 4.1.1 with:

(4) (xi) [[[[[[[[[[E_{i}, E_{j}], E_{k}], E_{l}], E_{k}], E_{j}], E_{k}], E_{l}], E_{k}], E_{j}], E_{k}] = 0

[figure: if; i; j; k; l; ×] ,

(xii) [[[[[E_{l}, E_{k}], E_{j}], E_{i}], E_{k}], E_{j}] = 2[[[[[E_{l}, E_{k}], E_{j}], E_{i}], E_{j}], E_{k}]

[figure: if; i; j; k; l; ⟹; ×] ,

(5) Relations of F_{i}’s defined as the same relations as (4).

5.3. An argument for affine G-type shall be different from the one for another affine type. Let (E_{0}, Π_{0}, p_{0}) = (E_{0} = ⊕_{i=1}^{4}Cα_{i}, Π_{0} = {α_{1}, α_{2}, α_{3}}, p_{0}) be the datum whose Dynkin diagram is:

Diagram 5.3.1.

[figure: 1; 2; 3; ×; <] .

Then the positive roots of G(E_{0}, Π_{0}, p_{0}) are:

Φ_{0,+} = {aα_{1} + bα_{2} + cα_{3} | (a, b, c) = (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 1), (1, 3, 1), (1, 3, 2), (1, 4, 2), (2, 4, 2), (0, 0, 1), (0, 1, 1), (0, 3, 2), (0, 2, 1), (0, 3, 1), (0, 1, 0)}.

Let (E, Π, p) = (E_{0}, Π_{0}, p_{0})^{(I)} (see 1.5). Then its Dynkin diagram is:

Diagram 5.3.2.

[figure: 0; 1; 2; 3; ×; <] .

Here the null root δ of G = G(E, Π, p) is given by δ = α_{0} + 2α_{1} + 4α_{2} + 2α_{3}. For i = 0, 2, 3, we define σ(i) : E → E by σ(i)(γ) = γ − \frac{2(γ,α_{i})}{(α_{i},α_{i})}α_{i}. For the set Φ_{+} of the positive roots of G, we put

\begin{matrix}Φ_{+}^{♭} = Φ_{+} ∪ ⋃_{i=0}^{3}(Φ_{+} + α_{i}), \\ Φ_{+}^{♯} = \{γ ∈ Φ_{+}^{♭}|(γ, γ) = 2(ρ, γ)\}. \end{matrix}

Then as a more precise fact than (1.2.4), it follows:

r_{+} = ⋃_{γ∈Φ_{+}^{♯}} r_{+,≤γ}  (5.3.1)

Since |2(ρ, δ)| = 12, it is clear that sufficiently large element of Φ_{+}^{♭} doesn’t belong to Φ_{+}^{♯}. By direct calculation, we can get:

Φ_{+}^{♯} = {α_{1}, 2α_{1}, α_{0}, α_{3}, α_{2}, α_{0} + 3α_{1} + 4α_{2} + α_{3}, α_{1} + 2α_{2}, α_{0} + α_{1} + α_{2} + α_{3}, 2α_{0} + 3α_{1} + 5α_{2} + 2α_{3}, α_{0} + 2α_{1} + 4α_{2} + 4α_{3}, α_{2} + 2α_{3}, α_{0} + 2α_{1} + 2α_{2} + α_{3}, 4α_{2} + α_{3}}.

Let r_{+}^{♯} be the ideal of Ñ^{+} generated by the relations (i), (ii), (iii) of Theorem 4.1.1 and

2[[[[[E_{0}, E_{1}], E_{2}], E_{3}], E_{1}], E_{2}] = 3[[[[[E_{0}, E_{1}], E_{2}], E_{3}], E_{2}], E_{1}] .  (5.3.2)

We also define the ideal r_{−}^{♯} of Ñ^{−} in the same way. By the criterion of Lemma 2.2.2, we can see r_{±}^{♯} ⊂ r_{±}. We can also see that r^{♯} = r_{−}^{♯} ⊕ r_{+}^{♯} is an ideal of G̃. Let G^{♯} = G̃/r^{♯}. Then we have a triangular decomposition G^{♯} = N^{♯,+} ⊕ H ⊕ N^{♯,−} where N^{♯,±} = Ñ^{±}/r_{±}^{♯}. Since the relations (iii) of Theorem 4.1.1 and their F_{i}’s version hold in G^{♯}, the automorphisms L_{i}^{♯} : G^{♯} → G^{♯} (i = 0, 2, 3) given by

L_{i}^{♯} = exp(adE_{i}) exp(ad(−\frac{2}{(α_{i}, α_{i})}F_{i})) exp(adE_{i})

are well-defined (see also [K1]). Let G^{♯} = H ⊕ (⊕_{α∈P_{+}∪P_{−}}G_{α}^{♯}) be the root space decomposition. Then we have L_{i}^{♯}(G_{α}^{♯}) = G_{σ(i)(α)}^{♯}. By Proposition 2.2.2, we have already had the automorphism L_{i} : G → G such that L_{i}(G_{α}) = G_{σ(i)(α)}. Clearly dim G_{α}^{♯} = dim G_{α} if α ∈ {α_{0}, α_{1}, 2α_{1}, α_{2}, α_{3}} Therefore, if β ∈ P_{+} is a minimal element under the order ≤ (see 1.2) among dim G_{β}^{♯} > dim G_{β}, then

β ∈ Φ_{+}^{♭} and (β, α_{0}) ≥ 0, (β, α_{2}) ≤ 0, (β, α_{3}) ≤ 0  (5.3.3)

because σ(i)(β) ≤ β (i = 0, 2, 3). The unique element satisfying (5.3.3) is α_{0} + 2α_{1} + 2α_{2} + α_{3}. However, using the relation (5.3.2), we see dim G_{α_{0}+2α_{1}+2α_{2}+α_{3}}^{♯} ≤ 1. Hence dim G_{α_{0}+2α_{1}+2α_{2}+α_{3}}^{♯} = dim G_{α_{0}+2α_{1}+2α_{2}+α_{3}}. Hence such β doesn’t exist. Hence G^{♯} is isomorphic to G.

By Proposition 2.2.1,we can get other Dynkin diagrams of G. Those are:

Diagram 5.3.3.

[figure: ×; <; ×; ×; ×; ×; ⟹; ×; ×; ×; ⟹] .

Theorem 5.3.1. Let (E, Π, p) be the data of which Dynkin diagrams existing in Diagram 5.2.3. Then defining relations of G(E, Π, p) are defined by adding the following relations to the ones of Theorem 4.1.1 with:

(4) (xiii) [[[E_{i}, E_{j}], [[E_{i}, E_{j}], [[E_{i}, E_{j}], E_{k}]]], E_{j}] = 0

[figure: if; i; j; k; ×; ×] ,

(xix) [E_{j}, [E_{k}, [E_{k}, [E_{j}, E_{i}]]]] = [E_{k}, [E_{j}, [E_{k}, [E_{j}, E_{i}]]]]

[figure: if; i; j; k; ×; ⟹] ,

(xx) 2[[[[[E_{l}, E_{k}], E_{j}], E_{i}], E_{k}], E_{j}] = 3[[[[[E_{l}, E_{k}], E_{j}], E_{i}], E_{j}], E_{k}]

[figure: l; k; j; i; if; ×; <] ,

(xxi) [[[[[[[[[[[[[[E_{i}, E_{j}], E_{k}], E_{l}], E_{k}], E_{j}], E_{k}], E_{l}], E_{k}], E_{j}], E_{k}], E_{l}], E_{k}], E_{j}], E_{k}] = 0

[figure: l; k; j; i; if; ×] ,

(5) Relations of F_{i}’s defined as the same relations as (4).

6. Quantization of relations 6.1. Let (E, Π, p) be a datum. Let C[[h]] denote the C-algebra of formal power series in h. In [Y1], we defined an h-adic topological Hopf superalgebra U_{h}(G) = U_{h}(G(E, Π, p)) in an abstract manner. For the terminologies of topological algebras, Hopf superalgebras etc., see [Y1]. Let Ũ_{h}^{♭}(B^{+})^{σ} be a non-topological C[[h]]-algebra defined with generators K_{λ} (λ ∈ ZΠ), E_{α} (α ∈ Π), σ and relations:

\begin{matrix}σ^{2} = 1, σK_{λ}σ = K_{λ}, σE_{α}σ = (−1)^{p(α)}E_{α}, \\ K_{0} = 1, K_{λ}K_{μ} = K_{λ+μ}, K_{λ}E_{α}K_{λ}^{−1} = exp((λ, α)h)E_{α}. \end{matrix}

It is easy to see that Ũ_{h}^{♭}(B^{+})^{σ} is a Hopf algebra with coproduct Δ, antipode S and counit ε such that

Δ(σ) = σ ⊗ σ, Δ(K_{λ}) = K_{λ} ⊗ K_{λ}, Δ(E_{α}) = E_{α} ⊗ 1 + K_{α}σ^{p(α)} ⊗ E_{α}
S(σ) = σ, S(K_{λ}) = K_{λ}^{−1}, S(E_{α}) = −K_{α}^{−1}σ^{p(α)}E_{α}
ε(σ) = 1, ε(K_{λ}) = 1, ε(E_{α}) = 0.

We note that Ũ_{h}^{♭}(B^{+})^{σ} is not a Hopf superalgebra but a Hopf algebra.

By [Y1], we have:

Lemma 6.1.1. (i)

E_{α(1)} ⋯ E_{α(r)}K_{λ}σ^{c} (α(j) ∈ Π, λ ∈ ZΠ, c ∈ {0, 1})

form a C[[h]]-basis of Ũ_{h}^{♭}(B^{+})^{σ}. In particular, as topological modules,

Ũ_{h}^{♭}(B^{+})^{σ} ≅ Ñ^{+} ⊗ C[[h]] [K_{λ}] ⊗ C[[h]]⟨σ⟩.  (6.1.1)

Here Ñ^{+}, C[[h]] [K_{λ}] and C[[h]]⟨σ⟩ denote the free algebra generated by E_{α} (α ∈ Π), the Laurent polynomial algebra in K_{α}^{±1} (α ∈ Π) and the group ring of {1, σ} respectively.

(ii) There is a symmetric Hopf pairing ⟨ , ⟩ : Ũ_{h}^{♭}(B^{+})^{σ} × Ũ_{h}^{♭}(B^{+})^{σ} → C[[h]] such that;

(a) Ñ^{+} and C[[h]] [K_{λ}] ⊗ C[[h]]⟨σ⟩ are orthogonal,

(b) ⟨E_{α}, E_{β}⟩ = δ_{α,β} (α, β ∈ Π),

(c) ⟨K_{λ}σ^{c}, K_{μ}σ^{d}⟩ = exp((λ, μ)h)(−1)^{cd}.

Remark 6.1.1 In [Y1], we introduced an another topological Hopf algebra Ũ′_{√\overline{h}}(b_{+})^{σ}. Then Ũ_{h}^{♭}(B^{+})^{σ} is given as the non-topological subalgebra of Ũ′_{√\overline{h}}(b_{+})^{σ} generated by E_{α}, K_{α}^{±} = exp(±\sqrt{h}H′_{α}) and σ.

Lemma 6.1.2. (See [Y1]) Let I^{+} = {X ∈ Ñ^{+}|⟨X, Y ⟩ = 0 (Y ∈ Ñ^{+})}. Then ker⟨ , ⟩ ≅ I^{+} ⊗ C[[h]] [K_{λ}] ⊗ C[[h]]⟨σ⟩ under (6.1.1). In particular, letting U_{h}^{♭}(B^{+})^{σ} = Ũ_{h}^{♭}(B^{+})^{σ}/ ker⟨ , ⟩ and N^{+} = Ñ^{+}/I^{+}, it follows that U_{h}^{♭}(B^{+})^{σ} ≅ N^{+} ⊗ C[[h]] [K_{λ}] ⊗ C[[h]]⟨σ⟩.

By [Y1] and Lemma 6.1.1, we have:

Theorem 6.1.1. For the datum (E, Π, p), there is an h-adic topological C[[h]]-Hopf superalgebra U_{h}(G) = U_{h}(G(E, Π, p)) with generators H_{λ} (λ ∈ E), E_{α}, F_{α} (α ∈ Π) with parities p(H_{λ}) = 0, p(E_{α}) = p(F_{α}) = p(α) satisfying following (a) and (b):

(a) In U_{h}(G), we have:

[H_{λ}, H_{μ}] = 0, [H_{λ}, E_{α}] = (λ, α)E_{α}, [H_{λ}, F_{α}] = −(λ, α)F_{α},

[E_{α}, F_{β}] = δ_{α,β} \frac{sinh(hH_{α})}{sinh(h)}. (6.1.3)

Put K_{λ} = exp(hH_{λ}). Then (U_{h}(G), Δ, S, ε) is a Hopf superalgebra such that

Δ(H_{λ}) = H_{λ} ⊗ 1 + 1 ⊗ H_{λ}, Δ(E_{α}) = E_{α} ⊗ 1 + K_{α} ⊗ E_{α}, Δ(F_{α}) = F_{α} ⊗ K_{α}^{−1} + 1 ⊗ F_{α}, S(H_{λ}) = −H_{λ}, S(E_{α}) = −K_{α}^{−1}E_{α}, S(F_{α}) = −F_{α}K_{α}, ε(H_{λ}) = ε(E_{α}) = ε(F_{α}) = 0. (6.1.4)

(b) Let C[[h]] [H] (resp. N^{+} or N^{−}) be the (non-topological) subalgebra of U_{h}(G) generated by H_{λ} (resp. E_{α} or F_{α}). Then C[[h]] [H] is the polynomial ring in H_{λ} ∈ H = E^{∗}. There are algebra isomorphisms Ñ^{+}/I^{+} → N^{+} (E_{α} → E_{α}) and N^{+} → N^{−} (E_{α} → F_{α}). There is a topological module isomorphism; U_{h}(G) ← (N^{−} ⊗ C[[h]] [H] ⊗ N^{+})^{∧} (Y QX ← Y ⊗ Q ⊗ X). (Here ( )^{∧} denotes completion.) In particular, U_{h}(G) is topologically free as an h-adic module.

6.2. Let U_{h}(G) = ⊕_{γ∈ZΠ}U_{h}(G)_{γ} be the weight space decomposition. (Here U_{h}(G)_{γ} = {X ∈ U_{h}(G)|[H_{λ}, X] = (λ, γ)X (λ ∈ E)}. Putting N_{γ}^{+} = N^{+} ∩ U_{h}(G)_{γ}, we have N^{+} = ⊕_{γ∈P_{+}}N_{γ}^{+}.

Lemma 6.2.1. There is an anti-homomorphism t : N^{+} → N^{+} (E_{α} → E_{α}) (α ∈ Π). Proof. Let S be the antipode of U_{h}^{♭}(B^{+})^{σ}. Define t by putting t(X) = (−1)^{∑_{i<j} p(α(i))p(α(j))} · exp(− ∑_{i<j}(α(i), α(j))h)K_{γ}S(X) for X ∈ N^{γ} with γ = ∑ α(i) (α(i) ∈ Π). By Lemma 6.1.2, we can easily check that t is the anti-homomorphism because S is so.

Q.E.D.

Let C((h)) be the quotient field of C[[h]]. Put C((h)) [H] = (C[[h]] [H])^{∧} ⊗_{C[[h]]} C((h)). Let C((h)) [K_{λ}] be the Laurent polynomial C((h))-algebra in K_{α}^{±1} (α ∈ Π). Then C((h)) [K_{λ}] = C[[h]] [K_{λ}]⊗_{C[[h]]} C((h)) and there is an epimorphism C((h)) [K_{λ}] ↪ C((h)) [H] (K_{λ} → exp(hH_{λ})) (λ ∈ ZΠ). We define e : U_{h}(G) → C((h)) [H] as the composition:

U_{h}(G) ≅ (N^{−} ⊗ C[[h]] [H] ⊗ N^{+})^{∧} ⟶^{ε⊗id⊗ε} C[[h]] [H] ↪ C((h)) [H] .

For γ ∈ ZΠ and T ∈ C((h)) [K_{λ}], denote the coefficient of K_{γ} of T by Q_{γ}(T) ∈ C((h)). denote the isomorphism N^{−} → N^{+} (F_{α} → E_{α}) by j. Put q = exp(h) and q^{a} = exp(ah). Let ⟨, ⟩ : N^{+} × N^{+} → C[[h]] be the non-degenerate symmetric pairing induced from ⟨, ⟩ of Lemma 6.1.1 (ii).

Lemma 6.2.2. Let γ = ∑_{α∈Π} l_{α}α ∈ P_{+} (l_{α} ∈ Z_{+}). Let X ∈ N_{γ}^{+} and Y ∈ N_{γ}^{−}. Then e(XY) ∈ C((h)) [K_{λ}] and we have:

Q_{γ}(e(XY)) = \frac{⟨t(X), j(Y)⟩}{(q − q^{−1})^{∑ l_{α}}} ∈ C((h)).  (6.2.1)

Proof. The first statement is clear. Describe γ ∈ P_{+} as two sums γ = ∑_{i=1}^{r} α(i) = ∑_{i=1}^{r} α′(i) (α(i), α′(i) ∈ Π). We compare two calculations of ⟨E_{α(r)}E_{α(r−1)} ⋯ E_{α(1)}, E_{α′(1)} ⋯ E_{α′(r−1)}E_{α′(r)}⟩ and E_{α(r)} ⋯ E_{α(r−1)}E_{α(r)}F_{α′(1)} ⋯ F_{α′(r−1)}F_{α′(r)}. By Δ(E_{α(i)}) = E_{α(i)} ⊗ 1 + K_{α(i)}σ^{p(α(i))} ⊗ E_{α(i)}, we can calculate:

⟨E_{α(r)}E_{α(r−1)} ⋯ E_{α(1)}, E_{α′(1)} ⋯ E_{α′(r−1)}E_{α′(r)}⟩

= ⟨E_{α(r)} ⊗ E_{α(r−1)} ⋯ E_{α(1)}, Δ(E_{α′(1)} ⋯ E_{α′(r−1)}E_{α′(r)})⟩

= ∑_{{1≤x≤r \atop α(r)=α′(x)}}⟨E_{α(r)} ⊗ E_{α(r−1)} ⋯ E_{α(1)},

K_{∑_{i=1}^{x−1} α′(i)}σ^{p(∑_{i=1}^{x−1} α′(i))}E_{α′(x)}K_{∑_{i=x+1}^{r} α′(i)}σ^{p(∑_{i=x+1}^{r} α′(i))} ⊗ E_{α′(1)} ⋯^{{x \atop ∨}} E_{α′(r)}⟩

= ∑_{{1≤x≤r \atop α(r)=α′(x)}}(−1)^{p(α(r))p(∑_{i=x+1}^{r} α′(i))}q^{−(α(r),∑_{i=x+1}^{r} α′(i))}⟨E_{α(r−1)} ⋯ E_{α(1)}, E_{α′(1)} ⋯^{{x \atop ∨}} E_{α′(r)}⟩

and E_{α(1)} ⋯ E_{α(r−1)}E_{α(r)}F_{α′(1)} ⋯ F_{α′(r)} = (−1)^{p(α(r))p(∑_{i=1}^{r} α′(i))}E_{α(1)} ⋯ E_{α(r−1)}F_{α′(1)} ⋯ F_{α′(r)}E_{α(r)}

+\frac{1}{(q−q^{−1})} ∑_{{1≤x≤r \atop α(r)=α′(x)}}(−1)^{p(α(r))p(∑_{i=1}^{x−1} α′(i))}

\{q^{−(α(r),∑_{i=x+1}^{r} α′(i))}E_{α(1)} ⋯ E_{α(r−1)}F_{α′(1)} ⋯^{{x \atop ∨}} F_{α′(r)}K_{α(r)}

−q^{(α(r),∑_{i=x+1}^{r} α′(i))}E_{α(1)} ⋯ E_{α(r−1)}F_{α′(1)} ⋯^{{x \atop ∨}} F_{α′(r)}K_{α(r)}^{−1}\}.

Since (−1)^{p(α(r))p(∑_{i=1}^{x−1} α′(i))} = (−1)^{p(α(r))+p(γ)p(α(r))}(−1)^{p(∑_{x+1}^{r} α′(i))}, we can reach (6.2.1) inductively.

Q.E.D.

Proposition 6.2.1. If X ∈ N^{+} satisfies

[X, F_{α}] = 0 for all α ∈ Π,  (6.2.2)

then X = 0. Proof. Since [N_{γ}^{+}, F_{α}] ⊂ N_{γ−α}^{+}, we may assume X ∈ N_{γ}^{+} for some γ ∈ P_{+}. By (6.2.2), we see that [X, F_{α(1)}F_{α(2)} ⋯ F_{α(r)}] = 0 for any {α(i)} with ∑_{i=1}^{r} α(i) = γ. Hence, by Lemma 6.2.2, it follow that ⟨X, X_{1}⟩ = 0 for any X_{1} ∈ N_{γ}^{+}. It is clear that the decomposition N^{+} = ⊕_{γ∈P_{+}}N_{γ}^{+} is orthogonal. Hence X ∈ ker⟨ , ⟩ whence X = 0.

Q.E.D.

6.3. For X_{α} ∈ U_{h}(G)_{α}, X_{β} ∈ U_{h}(G)_{β}, we put:

[[X_{α}, X_{β}]] = X_{α}X_{β} − (−1)^{p(α)p(β)}q^{−(α,β)}X_{β}X_{α}.

Let [x] = \frac{sinh(xh)}{sinh(h)} ∈ C[[h]]. By Proposition 6.2.1 and direct calculation, we have: Proposition 6.3.1. In N^{+}, we have:

(i) [E_{i}, E_{j}] = 0 if (α_{i}, α_{j}) = 0, (i ≠ j)

(ii) [E_{i}, E_{i}] = 0 if ^{i}^{}× , (iii) [[E_{i}, [[E_{i}, …, [[E_{i}, E_{j}]]…]]]] = 0 (E_{i} appear 1 − \frac{2(α_{i},α_{j})}{(α_{i},α_{i})}times)

if (α_{i}, α_{i}) ≠ 0 and p(α_{i})\frac{2(α_{j},α_{i})}{(α_{i},α_{i})} is even,

(iv) [[[[[E_{i}, E_{j}]], E_{k}]], E_{j}] = 0  if ^{i}  ^{j}  ^{k}^{−x}  ^{x} × ^{}× × (x ≠ 0),

(v) [[[[[E_{i}, E_{j}]], [[[[E_{i}, E_{j}]], E_{k}]]]], E_{j}] = 0 [figure: if; i; j; k; ×; ×; ⟸; ,]

(vi) [[[[[[[[[[[E_{i}, E_{j}]], E_{k}]], E_{l}]], E_{k}]], E_{j}]], E_{k}] = 0 [figure: if; i; j; k; l; ×; ×; ⟸; ,] (vii) (−1)^{p(α_{i})p(α_{k})}[(α_{i}, α_{k})][[[[E_{i}, E_{j}]], E_{k}]] = (−1)^{p(α_{i})p(α_{j})}[(α_{i}, α_{j})][[[[E_{i}, E_{k}]], E_{j}]]

[figure: if; j; k (ab ≠ 0); i; .; a; b; −a − b; .; .; and p(α_{i})p(α_{j}) + p(α_{i})p(α_{j}) + p(α_{i})p(α_{j}) ≡ 1,]

(viii) [[[[[[E_{i}, E_{j}]], [[E_{j}, E_{k}]]]], [[E_{j}, E_{l}]]]] = [x][[[[[[E_{i}, E_{j}]], [[E_{j}, E_{l}]]]], [[E_{j}, E_{k}]]]]

[figure: i; j; k; l; (x ≠ ±1, 0),; if; x + 1; ×; −1; −x]

(ix) [[[[E_{k}, [[E_{l}, [[E_{k}, E_{j},]]]]]], [[E_{k}, [[E_{l}, [[E_{k}, [[E_{j}, E_{i},]]]]]]]]]], E_{j}] = [2][[[[E_{k}, E_{j}]], [[[[E_{k}, [[E_{j}, E_{i}]]]], [[E_{k}, [[E_{l}, [[E_{k}, [[E_{j}, E_{i},]]]]]]]]]]]]

[figure: if; i; j; k; l; ⟹; ×; ,]

(x) [[E_{j}, [[E_{k}, [[E_{j}, [[E_{k}, E_{i}]]]]]]]] = [[E_{k}, [[E_{j}, [[E_{k}, [[E_{j}, E_{i}]]]]]]]]

[figure: if; i; j; k; ×; ×; ,]

(xi) [[[[[[[[[[[[[[[[[[[E_{i}, E_{j}]], E_{k}]], E_{l}]], E_{k}]], E_{j}]], E_{k}]], E_{l}]], E_{k}]], E_{j}]], E_{k}] = 0

[figure: if; i; j; k; l; ×] ,

(xii) [[[[[[[[[[E_{l}, E_{k}]], E_{j}]], E_{i}]], E_{k}]], E_{j}]] = [2][[[[[[[[[[E_{l}, E_{k}]], E_{j}]], E_{i}]], E_{j}]], E_{k}]]

[figure: if; i; j; k; l; ⟹; ×] ,

(xiii) [[[[[E_{i}, E_{j}]], [[[[E_{i}, E_{j}]], [[[[E_{i}, E_{j}]], E_{k}]]]]]], E_{j}] = 0

[figure: if; i; j; k; ×; ×] ,

(xix) [[E_{j}, [[E_{k}, [[E_{k}, [[E_{j}, E_{i}]]]]]]]] = [[E_{k}, [[E_{j}, [[E_{k}, [[E_{j}, E_{i}]]]]]]]]

[figure: if; i; j; k; ×; ⟹] ,

(xx) [2][[[[[[[[[[E_{l}, E_{k}]], E_{j}]], E_{i}]], E_{k}]], E_{j}]] = [3][[[[[[[[[[E_{l}, E_{k}]], E_{j}]], E_{i}]], E_{j}]], E_{k}]]

[figure: l; k; j; i; if; ×; <] ,

(xxi) [[[[[[[[[[[[[[[[[[[[[[[[[[[E_{i}, E_{j}]], E_{k}]], E_{l}]], E_{k}]], E_{j}]], E_{k}]], E_{l}]], E_{k}]], E_{j}]], E_{k}]], E_{l}]], E_{k}]], E_{j}]], E_{k}] = 0

[figure: l; k; j; i; if; ×] .

In 6.4, we shall describe how we calculate the relations (i)-(xix).

6.4. Let S be a C[[h]] or C((h))-superalgebra. For a ∈ C[[h]]^{×}, we put:

[X, Y]_{a} = XY − (−1)^{p(X)p(Y)}aY X (X, Y ∈ S).

Then we have

[[X, Y]_{a}, Z]_{b} = [X, [Y, Z]_{c}]_{abc^{−1}} + (−1)^{p(Y)p(Z)}c[[X, Z]_{bc^{−1}}, Y]_{ac^{−1}},  (6.4.1)

and

[X, [Y, Z]_{a}]_{b} = [[X, Y]_{c}, Z]_{abc^{−1}} + (−1)^{p(X)p(Y)}c[Y, [X, Z]_{bc^{−1}}]_{ac^{−1}} .  (6.4.2)

Hence, for U_{h}(G) and X_{ν} ∈ U_{h}(G)_{ν}, X_{μ} ∈ U_{h}(G)_{μ}, X_{η} ∈ U_{h}(G)_{η}, we have:

[[[[X_{ν}, X_{μ}]], X_{η}]] = [[X_{ν}, [[X_{μ}, X_{η}]]]] + (−1)^{p(μ)p(η)}q^{−(μ,η)}[[[X_{ν}, X_{η}]], X_{μ}]_{q^{(μ,η−ν)}},  (6.4.3)

and

[[X_{ν}, [[X_{μ}, X_{η}]]]] = [[[[X_{ν}, X_{μ}]], X_{η}]] + (−1)^{p(μ)p(ν)}q^{−(μ,ν)}[X_{μ}, [[X_{ν}, X_{η}]]]_{q^{(μ,ν−η)}}.  (6.4.4)

We can get the relations (i)-(xix) of Proposition 6.3.1 by Proposition 6.2.1 and direct calculation using (6.4.3-4). Here we only show how to get (ix) because the other relations can be also gotten similarly. We replace the letters i, j, k, l with 0, 1, 2, 3. We assume (α_{1}, α_{1}) = −2. Then the diagram can be rewritten as:

[figure: 0; 1; 2; 3; 2; 1; −2; ⟹; ×; .]

Put E_{...dcba} = [[ … [[E_{d}, [[E_{c}, [[E_{b}, E_{a}]]]]]]^{...}]]. Then (ix) is rewritten as:

−[[[[E_{2321}, E_{23210}]], E_{1}]] + (q + q^{−1})[[E_{21}[[E_{321}, E_{23210}]]]] = 0  (6.4.5)

We denote the LHS of (6.4.5) by X. Showing (6.2.2) is equivalent to showing [[X , F_{a}K_{a}^{−1}]] = 0 for all 0 ≤ a ≤ 3 where we put K_{a} = K_{α_{a}}. We note:

[[E_{α}, F_{β}K_{β}^{−1}]] = δ_{α,β} \frac{1 − K_{β}^{−2}}{q − q^{−1}} .  (6.4.6)

First we show [[X , F_{3}K_{3}^{−1}]] = 0. In following equations, LHS ∼ RHS mean LHS = a · RHS for some a ∈ C[[h]]^{×}. By (6.4.3) and (6.4.6), [[E_{321}, F_{3}K_{3}^{−1}]] ∼ [\frac{1−K_{3}^{−2}}{q−q^{−1}} , E_{21}]_{q^{(2+2)}} ∼ E_{21}. Hence

[[E_{2321}, F_{3}K_{3}^{−1}]]

∼ [[E_{2}, [[E_{321}, F_{3}K_{3}^{−1}]]]] by (6.4.3)

∼ [[E_{2}, E_{21}]]

= 0 by (ii) of Proposition 6.3.1.

Hence we also have [[E_{23210}, F_{3}K_{3}]] = 0. Hence we have [[[[[[E_{2321}, E_{23210}]], E_{1}]], F_{3}K_{3}^{−1}]] = 0. On the other hand,

[[[[E_{21}[[E_{321}, E_{2321}]]]], F_{3}K_{3}^{−1}]]
∼ [[E_{21}[[E_{21}, E_{23210}]]]] by (6.4.3) and E_{321}^{2} = 0
= 0 since E_{21}^{2} = 0.

Then we have [[X , F_{3}K_{3}]] = 0.

Next we show [[X , F_{2}K_{2}]] = 0. First we calculate: [[E_{2321}, F_{2}K_{2}^{−1}]] = [[[[E_{2}, E_{321}]], F_{2}K_{2}^{−1}]] = −q^{−1}[\frac{1−K_{2}^{−2}}{q−q^{−1}} , E_{321}]_{q^{(1+1)}} = E_{321},

[[E_{23210}, F_{2}K_{2}^{−1}]] = E_{3210},

[[[[E_{2321}, E_{23210}]], F_{2}K_{2}^{−1}]] = [[E_{2321}, E_{3210}]] +q^{−1}[E_{321}, E_{23210}]_{q^{(1−0)}} = −q^{−2}[E_{23210}, E_{321}]_{q^{(2+1)}} +q^{−1}[E_{321}, E_{23210}]_{q^{(1−0)}} (since E_{321}^{2} = 0) = (q + q^{−1})[[E_{321}, E_{23210}]],

[[[[E_{321}, E_{23210}]], F_{2}K_{2}^{−1}]] = 0 and [[E_{21}, F_{2}K_{2}^{−1}]] = E_{1}.

Using these, we have:

[[X , F_{2}K_{2}^{−1}]]

= −q[(q + q^{−1})[[E_{321}, E_{23210}]], E_{1}]_{q^{(−1−2)}} − (q + q^{−1})q^{−2}[E_{1}, [[E_{321}, E_{23210}]]]_{q^{(2+1)}} = 0

Similarly we can show [[X , F_{1}K_{1}^{−1}]] = [[X , F_{0}K_{0}^{−1}]] = 0. By Proposition 6.2.1, it follows that X = 0 ∈ N^{+}.

By similar calculation, we can get the relations (i)-(xix) of Proposition 6.3.1.

6.5. Let (A, Δ) be a cocommutative Hopf C-superalgebra. Let

P (A) = {x ∈ A|Δ(X) = X ⊗ 1 + 1 ⊗ X}.

Then P (A) is a Lie C-superalgebra with a bracket [ , ] given by [X, Y] = XY − (−1)^{p(X)p(Y)}Y X.

Let G be a Lie C-superalgebra and U(G) its universal enveloping superalgebra. Then U(G) is a cocommutative Hopf C-superalgebra with coproduct Δ such that Δ(X) = X ⊗ 1 + 1 ⊗ X for X ∈ U(G). It is known that:

Theorem 6.5.1(Milnor-Moor [MM]) Let CSH be the category of cocommutative Hopf Csuperalgebras. and SL the category of Lie C-superalgebras. Define morphisms P and U by P : CSH → SL (A → P (A)), U : SL → CSH (G → U(G)). Then PU = id_{CSH} and UP = id_{SL}.

As an immediate consequence of Theorem 6.5.1, we have:

Lemma 6.5.1 For a datum (E, Π, p), let G = G(E, Π, p) and U_{0}(G) a cocommutative Hopf Csuperalgebra defined by U_{0}(G) = U_{h}(G)/hU_{h}(G). Then there is an epimorphism

ϕ : U_{0}(G) → U(G) (H, E_{α}, F_{α} → H, E_{α}, F_{α}).

Proof. Let G_{0} = P (U_{0}(G)). By theorem 6.5.1, we have U_{0}(G) = U(G_{0}). Hence G_{0} should be a Lie C-superalgebra generated by H, E_{α}, F_{α}. By Theorem 6.1.1 (a), H, E_{α}, F_{α} satisfy (1.2.1-3). Since U_{h}(G) has the triangular decomposition by Theorem 6.1.1 (b), U_{0}(G) also has a triangular decomposition. In particular, H can be embedded into G_{0}(⊂ U_{0}(G)). By definition of the Kac-Moody Lie superalgebra G (see [K1]), we have epimorphism ϕ_{|G_{0}} : G_{0} → G (H, E_{α}, F_{α} → H, E_{α}, F_{α}). Hence we have ϕ.

Q.E.D.

6.6. Theorem 6.6.1 Let (E, Π, p) be a datum of affine type. Assume Ḡ(E, Π, p) = G(E, Π, p),i.e., G(E, Π, p) is not of ŝl(m|m)^{(i)} (i = 1, 2, 4). Put G = G(E, Π, p). Then the defining relations of U_{h}(G) are given by

(i) The relations of Theorem 6.1.1 (a), (ii) The relations of Proposition 6.3.1 (i)-(xix), (iii) The same relations of F_{α}’s as (ii).

Proof. By Theorem 6.1.1, Proposition 6.3.1, U_{h}(G) satisfies the relations (i)-(iii). Therefore, by Serre type theorems of Chapters 4 and 5, we have an epimorphism ψ : U(G) → U_{0}(G) (H, E_{α}, F_{α} → H, E_{α}, F_{α}). By Lemma 6.5.1, ψ should be the inverse map of ϕ. Then U_{0}(G) = U(G). The relations given by putting h = 0 on (i)-(iii) are the defining relations of U(G). Hence, by the topologically freedom of U_{h}(G), the relations (i)-(iii) should be the defining relations of U_{h}(G).

Q.E.D.

6.7 Lemma 6.7.1. Let U_{h}(Ġ) be the topologically free Hopf C[[h]]-superalgebra with generators {H, E_{α}, F_{α}} (α ∈ Π) such that

(1) {H, E_{α}, F_{α}} satisfy (6.1.3-4).

(2) The map H → U_{h}(Ġ) (H → H) is injective.

Then there is a Hopf superalgebra epimorphism

J̇ : U_{h}(Ġ) → U_{h}(G) (H, E_{α}, F_{α} → H, E_{α}, F_{α}).

Proof. By (2), the topological freedom of U_{h}(Ġ) and Theorem 6.5.1, C[[h]] [H] is embedded into U_{h}(Ġ). Let U_{h}(Ṅ^{+}), U_{h}(Ṅ^{−}) be non-topological subalgebras generated by E_{α}, F_{α} respectively. Put U_{0}(Ġ) = U_{h}(Ġ)/hU_{h}(Ġ) and U_{0}(Ṅ^{±}) = U_{h}(Ṅ^{±})/hU_{h}(Ṅ^{±}). By Minor-Moor’s Theorem 6.5.1, U_{0}(Ġ) (resp. U_{0}(Ṅ^{±})) is the universal enveloping algebra U(Ġ) (resp. U(Ṅ^{±})) of a Lie Csuperalgebra Ġ = P(U_{0}(Ġ)). (resp. Ṅ^{±} = P(U_{0}(Ṅ^{±})) By (2), H is embedded into Ġ. Hence we have the triangular decompositions Ġ = Ṅ^{−} ⊕ H ⊕ Ṅ^{+} and U(Ġ) = U(Ṅ^{−}) ⊗ U(H) ⊕ U(Ṅ^{+}). By the topological freedom of U_{h}(Ġ), we have the triangular decomposition U_{h}(Ġ) = U_{h}(Ṅ^{−}) ˆ⊗C[[h]] [H] ˆ⊗U_{h}(Ṅ^{+}).

Let İ^{+} = ker(Ñ^{+} → Ṅ^{+}). For γ ∈ P_{+}, put Ñ_{γ}^{+} (resp. İ_{γ}^{+}) = {X ∈ Ñ^{+} (resp İ) |[H_{λ}, X] = (λ, γ)X}. Then we have Ñ^{+} = ⊕_{γ∈P_{+}} Ñ_{γ}^{+} and İ^{+} = ⊕_{γ∈P_{+}} İ_{γ}^{+}. Keep notations in 6.2. Since İ^{+} is an ideal of Ñ^{+}, by the triangular decomposition of U_{h}(Ġ),

e([İ_{γ}^{+}, F_{α(1)} ⋯ F_{α(r)}]) = 0 for ∑ α(i) = γ (α(i) ∈ Π).

Hence, by a U_{h}(Ġ)-version of Lemma 6.2.2, since Ñ^{+} = ⊕_{γ∈P_{+}} Ñ_{γ}^{+} is orthogonal with respect to ⟨ , ⟩, we have İ_{γ}^{+} ⊂ I^{+} = ker⟨ , ⟩. Then we have the algebra epimorphism U_{h}(Ṅ^{+}) → N^{+}. (E_{α} → E_{α}).

Next we should show the existence of the epimorphism Ṅ^{−} → N^{−}. However a Hopf superalgebra (U_{h}(Ġ), sτ ∘ Δ, S^{−1}, ε) with generators {−H_{λ}, F_{α}, (−1)^{p(α)}E_{α}} also satisfies (1) and (2). (Here sτ(X ⊗ Y) = (−1)^{p(X)p(Y)}Y ⊗ X). Then the same argument can be applied for the subalgebra generated by F_{α}. Hence we can show the existence.

Eventually we get a C[[h]]-module surjective map:

J̇ : U_{h}(Ġ) = (Ṅ^{−} ⊗ C[[h]] [H] ⊗ Ṅ^{+})^{∧}U_{h}(G) = (N^{−} ⊗ C[[h]] [H] ⊗ N^{+})^{∧}.

Considering under the two triangular decompositions, it is clear that J̇ preserve product. Hence J̇ is the algebra epimorphism. Clearly J̇ is the Hopf superalgebra epimorphism.

Q.E.D.

7. Quantization of Weyl-group-type isomorphisms 7.1 Let (C, Δ, S, ε) be a topological Hopf C[[h]]-algebra. Define τ : C ˆ⊗C → C ˆ⊗C by τ(x⊗y) = y⊗x. Let Δ′ = τ ∘ Δ. Let C_{0} be a Hopf subalgebra of C. Let R = ∑ a_{i} ⊗ b_{i} be an invertible element of C_{0} ⊗ C_{0} satisfying:

RΔ(x)R^{−1} = Δ′(x), (7.1.1)

(Δ ⊗ I)(R) = R_{13}R_{23}, (I ⊗ Δ)(R) = R_{13}R_{12}. (7.1.2)

where R_{12} = R ⊗ I, R_{23} = I ⊗ R and R_{13} = ∑ a_{i} ⊗ I ⊗ b_{i}.

By Drinfeld[D2], we have known:

Proposition 7.1.1.(Drinfeld[D2])(i) R satisfies:

R_{12}R_{13}R_{23} = R_{23}R_{13}R_{13}, (7.1.3) (S ⊗ I)(R) = R^{−1} = (I ⊗ S^{−1})(R), (7.1.4) (ε ⊗ I)(R) = 1 = (I ⊗ ε)(R). (7.1.5)

(ii) For R = ∑ a_{i} ⊗ b_{i}, following equations hold in C:

∑ a_{i}S^{−2}(b_{i}) = ∑ S(a_{i})S^{−1}(b_{i}) = ∑ S^{2}(a_{i})b_{i}, (7.1.6) ∑ a_{i}S(b_{i}) = ∑ S^{−1}(a_{i})b_{i}. (7.1.7)

Let u_{4}, v_{4} ∈ C be the elements of (7.1.6), (7.1.7) respectively. Then u_{4}v_{4} = 1 = v_{4}u_{4}.

7.2. Proposition 7.2.1. Keep notations in 7.1. For the Hopf algebra C and the element R = ∑ a_{i} ⊗ b_{i} satisfying (7.1.1-2), there is an another Hopf algebra structure (C^{(R)}) = (C, Δ^{(R)}, S^{(R)}, ε) given by:

Δ^{(R)}(x) = RΔ(x)R^{−1}, S^{(R)}(x) = u_{4}^{−1}S(x)u_{4}.

Proof. First we show (I ⊗ Δ^{(R)}) ∘ Δ^{(R)} = (Δ^{(R)} ⊗ I) ∘ Δ^{(R)}. By (7.1.1-2), We have:

(I ⊗ Δ^{(R)}) ∘ Δ^{(R)}(x)
= R_{23}(I ⊗ Δ)(RΔ(x)R^{−1})R_{23}^{−1}
= R_{23}R_{13}R_{12}(I ⊗ Δ)(Δ(x))R_{12}^{−1}R_{13}^{−1}R_{23}^{−1}
= R_{12}R_{13}R_{23}(Δ ⊗ I)(Δ(x))R_{23}^{−1}R_{13}^{−1}R_{12}^{−1}
= (Δ^{(R)} ⊗ I) ∘ Δ^{(R)}(x).

Let m : C ⊗ C → C be the multiplication, which is defined by m(x ⊗ y) = xy. Next we show m ∘ (I ⊗ S^{(R)}) ∘ Δ^{(R)} = ε = m ∘ (S^{(R)} ⊗ I) ∘ Δ^{(R)}. By (7.1.4) and (7.1.6-7), for x ∈ C with Δ(x) = ∑ x_{i}^{(1)} ⊗ x_{i}^{(2)}, we have:

(I ⊗ S^{(R)}) ∘ Δ^{(R)}(x)
= ∑ m((1 ⊗ u_{4}^{−1})(I ⊗ S)(a_{i}x_{j}^{(1)}S(a_{l}) ⊗ b_{i}x_{j}^{(2)}b_{l})(1 ⊗ u_{4}))
= ∑ a_{i}x_{j}^{(1)}S(a_{l}) · a_{y}S(b_{l})S(b_{i})S(x_{j}^{(2)})S(b_{i})u_{4}
= ∑ a_{i}x_{j}^{(1)}S(x_{j}^{(2)})S(b_{i})u_{4} since (S ⊗ I)(R)R = 1
= ε(x) ∑ a_{i}S(b_{i})u_{4} = ε(x)v_{4}u_{4} = ε(x)

and

(S^{(R)} ⊗ I) ∘ Δ^{(R)}(x)
= ∑ m((u_{4}^{−1} ⊗ 1)(S ⊗ I)(a_{i}x_{j}^{(1)}S(a_{l}) ⊗ b_{i}x_{j}^{(2)}b_{l})(u_{4} ⊗ 1))
= ∑ u_{4}^{−1}S^{2}(a_{l})S(x_{j}^{(1)})S(a_{i})S(a_{y})S^{−1}(b_{y}) · b_{i}x_{j}^{(2)}b_{l}
= ∑ u_{4}^{−1}S^{2}(a_{l})S(x_{j}^{(1)})x_{j}^{(2)}b_{l} since (I ⊗ S^{−1})(R)R = 1
= ε(x)u_{4}^{−1} ∑ S^{2}(a_{l})b_{l} = ε(x)u_{4}^{−1}u_{4} = ε(x).

To show other formulae of the axiom of the Hopf algebra are easy.

Q.E.D.

7.3. Let (E, Π, p) be a datum and G = G(E, Π, p). Let U_{h}(G) be a topological Hopf C[[h]]- superalgebra introduced in 6.1. Put U_{h}(G)^{σ} = U_{h}(G) ⊗_{C[[h]]} C[[h]]⟨σ⟩. Then U_{h}(G)^{σ} is an algebra with a formula σXσ = (−1)^{p(X)}X (X ∈ U_{h}(G)). By [Y1], (U_{h}(G)^{σ}, Δ, S, ε) is a Hopf algebra such that

Δ(H) = H ⊗ 1 + 1 ⊗ H, Δ(E_{α}) = E_{α} ⊗ 1 + K_{α}σ^{p(α)} ⊗ E_{α},
Δ(F_{α}) = F_{α} ⊗ K_{α}^{−1} + σ^{p(α)} ⊗ F_{α},
S(H) = −H, S(E_{α}) = −K_{α}^{−1}σ^{p(α)}E_{α}, S(F_{α}) = −σ^{p(α)}F_{α}K_{α}
ε(H) = ε(E_{α}) = ε(F_{α}) = 0.

Put U_{h}(H)^{σ} = C[[h]][H] ⊗ C[[h]]⟨σ⟩. Then U_{h}(H)^{σ} is a Hopf subalgebra of U_{h}(G)^{σ}. Put t_{0} = ∑ H_{δ_{i}} ⊗ H_{δ_{i}} ∈ H ⊗ H where {δ_{i}} is a C-basis of H such that (δ_{i}, δ_{j}) = δ_{ij}. Then, by the quantum double construction (see [D] (also [Y1])),

R_{T} = \frac{1}{2}( ∑_{c,d=0,1}(−1)^{cd}σ^{c} ⊗ σ^{d}) · exp(−ht_{0}) ∈ U_{h}(H)^{σ} ⊗ U_{h}(H)^{σ}

satisfies (7.1.1-2). Clearly R_{T}^{−1} also satisfies (7.1.1-2).

For t ∈ C[[h]] and n > 0, we put \{n\}_{t} = \frac{t^{n}−1}{t−1}, \{n\}_{t}! = \{n\}_{t}\{n − 1\}_{t} ⋯ \{1\}_{t} and

\left\{{n \atop m}\right\}_{t} = \{ \begin{matrix}\frac{\{n\}_{t}!}{\{m\}_{t}!\{n−m\}_{t}!} & if n ≤ m ≤ 0, \\ 0 & otherwise. \end{matrix}

For u ∈ hU_{h}(G)^{σ}, put e(u, t) = ∑_{n=0}^{∞} \frac{u^{n}}{\{n\}_{t}!}. It is easy to show that

e(−u, t^{−1}) = e(u, t)^{−1}, (7.3.1) e(u, t)Xe(u, t)^{−1} = ∑_{n=0}^{∞} \frac{1}{\{n\}_{t}!}ad_{t^{n−1}}(u)ad_{t^{n−2}}(u) ⋯ ad_{1}(u)(X) (7.3.2)

where ad_{x}(u)(X) = [u, X]_{−,x} = uX − xXu.

For α ∈ Π, let U_{h}(G^{(α)})^{σ} be a topological subalgebra of U_{h}(G)^{σ} generated by U_{h}(H)^{σ} and E_{α}, F_{α}. By the quantum double construction, we see that

R_{α} = e(−(q − q^{−1})E_{α} ⊗ F_{α}σ^{p(α)}, (−1)^{p(α)}q^{(α,α)}) · R_{T} ∈ U_{h}(G^{(α)})^{σ} ⊗ U_{h}(G^{(α)})^{σ}

satisfies (7.1.1-2). Let (U_{h}(G)^{σ})^{(α)} = (U_{h}(G)^{σ}, Δ^{(α)}, S^{(α)}, ε) be an another Hopf algebra defined as ((U_{h}(G)^{σ})^{(R_{α})})^{(R_{T}^{−1})}. Put

R̂_{α} = e(−(q − q^{−1})σ^{p(α)}K_{α}^{−1}E_{α} ⊗ F_{α}K_{α}, (−1)^{p(α)}q^{(α,α)}).

Then we get R̂_{α} = R_{T}^{−1}R_{α}. Hence

Δ^{(α)}(X) = R̂_{α}Δ(X)R̂_{α}^{−1} (X ∈ U_{h}(G)^{σ}).

Proposition 7.3.1. For α, β ∈ Π. Put E_{β+sα}^{∨} = [[...[[[[E_{β}, E_{α}]], E_{α}]] … E_{α}]], F_{β+sα}^{∨} = [[...[[[[F_{β}, F_{α}]], F_{α}]] … F_{α}]] (E_{α}, F_{α} appears s-times).

(i) Δ^{(α)}(E_{α}K_{α}^{−1}) = E_{α}K_{α}^{−1} ⊗ K_{α} + σ^{p(α)} ⊗ E_{α}K_{α}^{−1}, Δ^{(α)}(K_{α}F_{α}) = K_{α}F_{α} ⊗ 1 + σ^{p(α)}K_{α}^{−1} ⊗ K_{α}F_{α}.

(ii) Assume (α, α) ≠ 0. For β ∈ Π, assume r = r_{(α,β)} = −\frac{2(α,β)}{(α,α)} ∈ Z_{+} and p(α) · r is even. Then

\begin{matrix}Δ^{(α)}(E_{β+rα}^{∨}) = E_{β+rα}^{∨} ⊗ 1 + K_{β+rα}σ^{p(β+rα)} ⊗ E_{β+rα}^{∨}, \\ Δ^{(α)}(F_{β+rα}^{∨}) = F_{β+rα}^{∨} ⊗ K_{β+rα}^{−1} + σ^{p(β+rα)} ⊗ F_{β+rα}^{∨}. \end{matrix}

(iii) Assume (α, α) = 0 and (α, β) ≠ 0. Then

\begin{matrix}Δ^{(α)}(E_{β+α}^{∨}) = E_{β+α}^{∨} ⊗ 1 + K_{β+α}σ^{p(β+α)} ⊗ E_{β+α}^{∨}, \\ Δ^{(α)}(F_{β+α}^{∨}) = F_{β+α}^{∨} ⊗ K_{β+α}^{−1} + σ^{p(β+α)} ⊗ F_{β+α}^{∨}. \end{matrix}

(iv) Δ^{(α)}(H) = H ⊗ 1 + 1 ⊗ H, Δ^{(α)}(σ) = σ ⊗ σ. Proof. Here we calculate Δ^{(α)}(E_{β+rα}^{∨}) of (ii). Put t_{α} = (−1)^{p(α)}q^{(α,α)} and t_{α,β} = (−1)^{p(α)p(β)}q^{(α,β)}. By direct calculation, we have:

Δ(E_{β+uα}^{∨}) = E_{β+uα}^{∨} ⊗ 1

+ ∑_{s=0}^{u} \left\{{u \atop s}\right\}_{t_{α}} ∏_{k=1}^{u−s}(t_{α,β}^{−1} − t_{α,β}^{−1}t_{α}^{k−u})E_{α}^{u−s}K_{β+sα}σ^{p(β+sα)} ⊗ E_{β+sα}^{∨}.

Hence, for r of (ii),

Δ(E_{β+rα}^{∨}) = E_{β+rα}^{∨} ⊗ 1

+ ∑_{s=0}^{r} \left\{{r \atop s}\right\}_{t_{α}}t_{α,β}^{r−s} ∏_{k=1}^{r−s}(1 − t_{α}^{k})E_{α}^{r−s}K_{β+sα}σ^{p(β+sα)} ⊗ E_{β+sα}^{∨}.

Put X = −σ^{p(α)}K_{α}^{−1} ⊗ F_{α}K_{α}. Then

[−X, E_{α}^{s}K_{β+(r−s)α}σ^{p(β+(r−s)α)} ⊗ E_{β+(r−s)α}^{∨}]_{−,t^{−s}}

= \{ \begin{matrix}−\frac{t_{α,β}}{q−q^{−1}} t_{α}^{−s} \frac{(t_{α}^{r−s}−1)(t_{α}^{s+1}−1)}{t_{α}−1}E_{α}^{s+1}K_{β+(r−s−1)α}σ^{p(β+(r−s−1)α)} ⊗ E_{β+(r−s−1)α}^{∨} \\ if r > s, \\ 0 if r = s. \end{matrix}

Hence

ad_{t_{α}^{−(s−1)}}(X)ad_{t_{α}^{−(s−2)}}(X) ⋯ ad_{1}(X)(K_{β+rα}σ^{p(β+rα)} ⊗ E_{β+rα}^{∨})

= \{ \begin{matrix}= (\frac{t_{α,β}}{q−q^{−1}} )^{s}t_{α}^{− \frac{s(s−1)}{2}} (t_{α} − 1)^{s}\left\{{r \atop s}\right\}_{t_{α}}E_{α}^{s}K_{β+(r−s)α}σ^{p(β+(r−s)α)} ⊗ E_{β+(r−s)α}^{∨} \\ if r ≥ s, \\ 0 if r < s. \end{matrix}

Hence, by (7.3.1-2),

R̂_{α}^{−1}(K_{β+rα}σ^{p(β+rα)} ⊗ E_{β+rα}^{∨})R̂_{α}

= ∑_{s=0}^{r}(−t_{α})^{s}(t − 1)^{s} \frac{\{r\}_{t_{α}}!}{\{r − s\}_{t_{α}}!}E_{α}^{s}K_{β+(r−s)α}σ^{p(β+(r−s)α)} ⊗ E_{β+(r−s)α}^{∨} .

On the other hand, we can easily show R̂_{α}^{−1}(E_{β+rα}^{∨} ⊗ 1)R̂_{α} = E_{β+rα}^{∨} ⊗ 1. Then we get:

R̂_{α}^{−1}(E_{β+rα}^{∨} ⊗ 1 + K_{β+rα}σ^{p(β+rα)} ⊗ E_{β+rα}^{∨})R̂_{α} = Δ(E_{β+rα}^{∨}) .

We can show other formulae similarly or easily.

Q.E.D.

By [Y1], we see:

Lemma 7.3.1. For α ∈ Π, U_{h}(G) has an another Hopf superalgebra structure U_{h}(G)^{(α)} = (U_{h}(G), Δ_{s}^{(α)}) with coproduct Δ_{s}^{(α)} satisfies formulae given by eliminating σ^{p(·)} in the formulae of Δ^{(α)} of Proposition 7.3.1 (i)-(iv).

7.4. Lemma 7.4.1. Keep notation in 7.3. Let α ∈ Π.

(i) Assume (α, α) ≠ 0. Assume r = r_{α,β} ∈ Z_{+} and p(α)r ∈ 2Z for any β ∈ Π \ {α}. Define σ_{α} : H → H by σ_{α}(H_{λ}) = H_{λ− \frac{2(α,λ)}{(α,α)} α}. Let x_{β}, y_{β} ∈ C[[h]]^{×} (β ∈ Π) be such that

x_{β}y_{β} = \{ \begin{matrix}(−1)^{p(β)} & (β = α), \\ (−1)^{r_{α,β}}q^{−(r_{α,β}+1)(α,β)}(\frac{1−t_{α}^{−1}}{q−q^{−1}} )^{r_{α,β}}(\{r_{α,β}\}_{t_{α}^{−1}}!)^{2} & (β ≠ α, (α, β) ≠ 0), \\ 1 & (β ≠ α, (α, β) = 0). \end{matrix}

Put H′_{λ} = H_{σ_{α}(λ)}, E′_{α} = x_{α}^{−1}F_{α}K_{α}, F′_{α} = y_{α}^{−1}K_{α}^{−1}E_{α}, E′_{β} = x_{β}^{−1}E_{β+r_{α,β}α}^{∨}, F′_{β} = y_{β}^{−1}F_{β+r_{α,β}α}^{∨} (β ∈ Π \ {α}).

Then H′_{λ}, E′_{β}, F′_{β} satisfy (6.1.3).

(ii) Assume (α, α) = 0. For β ∈ Π \ {α}, put

r_{α,β} = \{ \begin{matrix}1 & (α, β) ≠ 0 \\ 0 & (α, β) = 0 \end{matrix}

Let Π′ = {α′ = −α, β′ = α + β (β ∈ Π, (α, β) ≠ 0), γ′ = γ (γ ∈ Π \ {α}, (γ, α) = 0)}. Let σ_{α} : (E, Π, p) → (E, Π′, p) by σ_{α}(H) = H. (In particular,

σ_{α}(H_{β}) = \{ \begin{matrix}H_{−α′} & (β = α), \\ H_{β′+r_{α,β}α′} & (β ≠ α, (α, β) ≠ 0).) \end{matrix}

Let x_{β}, y_{β} ∈ C[[h]]^{×} (β ∈ Π) be such that

x_{β}y_{β} = \{ \begin{matrix}−1 & (β = α), \\ t_{α,β}^{−1} \frac{q^{(α,β)}−q^{−(α,β)}}{q−q^{−1}} & (β ≠ α, (α, β) ≠ 0), \\ 1 & (β ≠ α, (α, β) = 0). \end{matrix}

Put E′_{α} = x_{α}^{−1}F_{α}K_{α}, F′_{α} = y_{α}^{−1}K_{α}^{−1}E_{α}, E′_{β} = x_{β}^{−1}E_{β+r_{α,β}α}^{∨}, F′_{β} = y_{β}^{−1}F_{β+r_{α,β}α}^{∨} (β ∈ Π \ {α}).

Then H, E′_{β}, F′_{β} satisfy (6.1.3) for (E, Π′, p). Proof. Here we show how to calculate

[E_{β+r_{α,β}α}^{∨}, F_{β+r_{α,β}α}^{∨}] = x_{β}y_{β} \frac{sinh(hH_{β+r_{α,β}α})}{sinh(h)}.  (7.4.1)

Put \{k; β\}_{α} = \frac{q^{−(α,β)}(1−t_{α}^{−k})(1−t_{α,β}^{2}t_{α}^{k−1})}{(q−q^{−1})(1−t_{α}^{−1})} and \{k; β\}_{α}! = ∏_{v=1}^{k}\{v; β\}_{α}. First we show:

\begin{matrix}[E_{α}, F_{β+kα}^{∨}] = −t_{α,β}^{−1}q^{−(k−1)(α,α)}\{k; β\}_{α}F_{β+(k−1)α}^{∨}K_{α} , \\ [E_{β+kα}^{∨}, F_{α}] = (−1)^{(k−1)p(α)}\{k; β\}_{α}K_{α}^{−1}E_{β+(k−1)α}^{∨} . \end{matrix}

Then, by induction on k, we can show:

[E_{β+kα}^{∨}, F_{β+(k−1)α}^{∨}]
= (−1)^{k(1+p(α)p(β))}q^{− \frac{(k−1)(k−2)}{2} (α,α)}q^{(−k+1)(α,β)}\{k; β\}_{α}!E_{α}K_{β+(k−1)α},

[E_{β+(k−1)α}^{∨}, F_{β+kα}^{∨}]
= (−1)^{(k−1)(1+p(α)+p(α)p(β))}q^{− \frac{k(k−1)}{2} (α,α)}q^{−k(α,β)}\{k; β\}_{α}!K_{β+(k−1)α}^{−1}F_{α}

and

[E_{β+kα}^{∨}, F_{β+kα}^{∨}]

= (−1)^{k}t_{α,β}^{−k}q^{− \frac{k(k−1)}{2} (α,α)}\{k; β\}_{α}!\frac{sinh(hH_{β+kα})}{sinh(h)}.

Substituting r_{α,β} for k, we get (7.3.1).

We can show other formulae similarly or easily.

Q.E.D.

7.5. Proposition 7.5.1. Keep notations in 7.4.

(i) Let Π^{σ_{α}} = Π if (α, α) ≠ 0 and let Π^{σ_{α}} = Π′ if (α, α) = 0. Put U_{h}(G^{σ_{α}}) = U_{h}(G(E, Π^{σ_{α}}, p)). Then there is an isomorphism L_{α} : U_{h}(G) → U_{h}(G^{σ_{α}}) such that

L_{α}(H) = σ_{α}(H), L_{α}(E_{β}) = E′_{β}, L_{α}(F_{β}) = F′_{β}.  (7.5.1)

(ii)

Δ(L_{α}(X)) = R̂_{α}^{−1}(L_{α} ⊗ L_{α}Δ(X))R̂_{α} (X ∈ U_{h}(G)^{σ}).  (7.5.2)

Proof. (i) By Lemma 6.7.1, Lemma 7.3.1 and Lemma 7.4.1, there is an epimorphism L′_{α} : U_{h}(G) → U_{h}(G^{σ_{α}}) satisfying (7.5.1). Let L_{α} : U_{h}(G) → U_{h}(G^{σ_{α}}) denote L′_{α} defined by changing U_{h}(G) and U_{h}(G^{σ_{α}}). (Keep notations in the proof of Lemma 6.5.1.) Since L′_{α}L_{α}_{|H} = id_{H} and (resp. L_{α}L′_{α}_{|H} = id_{H}), L′_{α}L_{α} (resp. L_{α}L′_{α}) induce an automorphism of G_{0} (resp. G_{0}^{σ_{α}}) as well as an automorphism of U_{0}(G_{0}) (resp. U_{0}(G_{0}^{σ_{α}})). Hence L_{α} induce an isomorphism U_{0}(G_{0}) → U_{0}(G_{0}^{σ_{α}}). Hence by topological freedom of U_{h}(G) and U_{h}(G^{σ_{α}}), L_{α} is an isomorphism. (ii) (7.5.2) is clear from the formulae in Proposition 7.3.1 (i)-(iii).

Q.E.D.

By direct calculation, we have:

Lemma 7.5.1. L_{α}^{−1} : U_{h}(G) → U_{h}(G^{σ_{α}}) satisfies (Here we let the region of definition (resp. values) of L_{α}^{−1} be (E, Π, p) (resp. (E, Π^{σ_{α}}, p))). For α, β ∈ Π. Put E_{β+sα} = [[E_{α} … [[E_{α}, [[E_{α}, E_{β}]]]]...]], F_{β+sα} = [[F_{α} … [[F_{α}, [[F_{α}, F_{β}]]]]...]], (E_{α}, F_{α} appears s-times).

Put H′′_{λ} = H_{σ_{α}(λ)}, E′′_{α} = ẋ_{α}^{−1}K_{α}^{−1}F_{α}, F′′_{α} = ẏ_{α}^{−1}E_{α}K_{α}, E′′_{β} = ẋ_{β}^{−1}E_{β+r_{α,β}α}, F′′_{β} = ẏ_{β}^{−1}F_{β+r_{α,β}α} (β ∈ Π \ {α}). Here we define ẋ_{β}, ẏ_{β} ∈ C[[h]]^{×} by:

x_{α}ẏ_{α} = y_{α}ẋ_{α} = 1,
y_{α}^{r_{α,β}}y_{β}ẏ_{β} = (−1)^{r_{α,β}}(−1)^{(p(α)+p(α)p(β))r_{α,β}}{r_{α,β}; β}_{α}!,
x_{α}^{r_{α,β}}x_{β}ẋ_{β} = (−1)^{r_{α,β}}(−1)^{p(α)p(β)r_{α,β}}q^{r_{α,β}(α,α)}{r_{α,β}; β}_{α}! (β ≠ α, (α, β) ≠ 0),
x_{β}ẋ_{β} = y_{β}ẋ_{β} = 1 (β ≠ α, (α, β) = 0).

(Here x_{β}, y_{β} ∈ C[[h]]^{×} have been defined in Lemma 7.4.1 for L_{α} : U_{h}(G^{σ_{α}}) → U_{h}(G).)

7.6. As an immediate consequence of Proposition 7.5.1, we have:

Proposition 7.6.1. (See also [KT].) Let (E, Π, p)’s be the data of affine type. For the isomorphisms L_{i} defined for G(E, Π, p)’s in §2, there are isomorphisms T_{i}’s of U_{h}(G(E, Π, p))’s such that T_{i} → L_{i} : U_{0}(G(E, Π, p)) → U_{0}(G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)})) (h → 0).

8. On U_{h}(ŝl(m|m))^{(i)} (i = 1, 2, 4). In this chapter, we use Beck’s method [B]. 8.1. In 8.1, let (E, Π, p) be of Diagram 1.6.2 and assume N ≥ 4. Let W be the Weyl group defined in 2.6 associated to (E^{†}, Π^{†}). Let W_{0} be a subgroup of W generated by {σ(i), (1 ≤ i ≤ n)}. Let

ω_{j}^{∨} ∈ ⊕_{i=1}^{n}Cα_{i}^{†} (1 ≤ j ≤ n) be such that \frac{2((α_{i}^{†},ω_{j}^{∨}))}{((α_{i}^{†},α_{i}^{†}))} = δ_{ij} (1 ≤ i ≤ n). Put P^{∨} = ⊕Zω_{i}^{∨}. Define

W̄ = W_{0}|× P^{∨} by (s, x)(s′, x′) = (ss′, s^{′−1}(x) + x′). We know that there is a certain subgroup T of Dynkin diagram automorphism of (E^{†}, Π^{†}) such that W̄ ≅ T |× W_{0} (τσ(i)τ^{−1} = σ(τ(i)) (τ ∈ T )). If W is of type A_{N−1}^{(1)}, then T ≅ Z/NZ. For the datum (E = (⊕_{i=1}^{N}Cε̄_{i}) ⊕ Cδ ⊕ Cλ_{0}, Π = {α_{i}}, p) and τ ∈ T , define the datum (E^{τ} = (⊕_{i=1}^{N}Cε̄_{i}^{τ}) ⊕ Cδ ⊕ Cλ_{0}, Π^{τ} = {α_{i}^{τ}}, p^{τ}) by

(i) The Dynkin diagram of (E^{τ}, Π^{τ}, p^{τ}) is the same type as the one of (E, Π, p). (ii) p^{τ}(α_{i}) = p(α_{τ^{−1}(i)}). (iii) (ε̄_{i}^{τ}, ε̄_{j}^{τ}) = (ε̄_{τ^{−1}(i)}, ε̄_{τ^{−1}(j)}) (Here we consider τ^{−1}(i) under modN).

For w ∈ W̄ and an reduced expression w = τσ(i_{1}) ⋯ σ(i_{r}), put (E^{w}, Π^{w}, p^{w}) = ( (( (E^{σ(i_{r})}) ⋯)^{σ(i_{1})})^{τ}, (( (Π^{σ(i_{r})}) ⋯)^{σ(i_{1})})^{τ}, (( (p^{σ(i_{r})}) ⋯)^{σ(i_{1})})^{τ}). Clearly (E^{w}, Π^{w}, p^{w}) doesn’t depend on reduced expressions.

Let U_{h}(G)′ be the subalgebra of U_{h}(G) generated by {H_{α_{i}}, E_{i}, F_{i} (0 ≤ i ≤ n)}. By Proposition 7.5.1, Lemma 7.5.1 and direct calculation, we have:

Lemma 8.1.1. Assume that (E, Π, p) is type A_{N−1}^{(1)}. Keep notations in 2.3-5. Let i ∈ {0, 1, ..., N − 1}(= Z/NZ). Put K_{i} = K_{α_{i}}.

(i) There are isomorphisms T_{i} : U_{h}(G(E, Π, p))′ → U_{h}(G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}))′ such that (We put p′ = p^{σ(i)}.)

T_{i}E_{i} = −d̄′_{i+1}F_{i}K_{i}, T_{i}F_{i} = −d̄′_{i}K_{i}^{−1}E_{i},
T_{i}E_{i−1} = q^{−d̄′_{i}} d̄′_{i}[[E_{i−1}, E_{i}]], T_{i}E_{i+1} = q^{−d̄′_{i+1}}(−1)^{p′(α_{i})p′(α_{i+1})}d̄′_{i+1}[[E_{i+1}, E_{i}]],
T_{i}F_{i−1} = −(−1)^{p′(α_{i})p′(α_{i−1})}[[F_{i−1}, F_{i}]], T_{i}F_{i+1} = −[[F_{i+1}, F_{i}]].

T_{i}^{−1} : U_{h}(G(E, Π, p))′ → U_{h}(G(E^{σ(i)}, Π^{σ(i)}, p^{σ(i)}))′ is given by:

T_{i}^{−1}E_{i} = −d̄′_{i+1}K_{i}^{−1}F_{i}, T_{i}^{−1}F_{i} = −d̄′_{i}E_{i}K_{i},

T_{i}^{−1}E_{i−1} = q^{−d̄′_{i}}(−1)^{p′(α_{i})p′(α_{i−1})}d̄′_{i}[[E_{i}, E_{i−1}]], T_{i}^{−1}E_{i+1} = q^{−d̄′_{i+1}} d̄′_{i+1}[[E_{i}, E_{i+1}]],
T_{i}^{−1}F_{i−1} = −[[F_{i}, F_{i−1}]], T_{i}^{−1}F_{i+1} = −(−1)^{p′(α_{i})p′(α_{i+1})}[[F_{i}, F_{i+1}]].

For τ ∈ T , there is an isomorphism T_{τ} : U_{h}(G(E, Π, p))′ → U_{h}(G(E^{τ}, Π^{τ}, p^{τ}))′ such that T_{τ}(H_{α_{i}}) = H_{α_{τ(i)}}, T_{τ}(E_{i}) = E_{τ(i)}, T_{τ}(F_{i}) = F_{τ(i)}. (ii) T_{i}’s satisfy Braid relation:

T_{i}T_{j} = T_{j}T_{i} ((α_{i}, α_{j}) = 0), T_{i}T_{j}T_{i} = T_{j}T_{i}T_{j} (|(α_{i}, α_{j})| = 1).

It also hold that T_{τ}T_{i}T_{τ}^{−1} = T_{τ(i)}. (iii) By (ii), putting T_{w} = T_{τ}T_{i_{1}} ⋯ T_{i_{r}} for w ∈ W̄ whose reduced expression is w = τσ(i_{1}) ⋯ σ(i_{r}), T_{w} is well-defined. Moreover we have:

T_{w}(E_{i}) = E_{j}, T_{w}(F_{i}) = F_{j} if w(α_{i}) = α_{j}.

There is an C-anti-automorphism Ω such that

Ω(E_{i}) = d̄_{i+1}F_{i}, Ω(E_{i}) = d̄_{i+1}F_{i}, Ω(H) = H, Ω(h) = −h .

Moreover ΩT_{w} = T_{w}Ω (w ∈ W̄). 8.2. Put T_{ω_{i}} = T_{ω_{i}^{∨}}. For 1 ≤ i ≤ n, k > 0 and s ∈ Z, let

ψ̄_{ik}^{(s)} = K_{δ}^{− \frac{k}{2}} q^{−(α_{i},α_{i})}[[T_{ω_{i}}^{s}(E_{i}), T_{ω_{i}}^{k+s}(K_{i}^{−1}F_{i})]] .  (8.1.1)

Put Q_{ij,k} = \frac{q^{k(α_{i},α_{j})}−q^{−k(α_{i},α_{j})}}{q−q^{−1}} and Ċ_{ij} = q^{(α_{i},α_{j})}K_{δ}^{− \frac{1}{2}} . By [B], we have: Lemma 8.2.1. (i) Kδ^{\frac{1}{k}} ψ̄_{ik}^{(s)} ∈ N_{+} if s ≤ 0 and k + s > 0. (ii) Assume p(α_{i}) = 0. Let r > 0 and m ∈ Z. Then ψ̄_{ir}^{(s)} = ψ̄_{ir}^{(s′)} (s, s′ ∈ Z) and

[ψ̄_{ir}^{(s)}, T_{ω_{i}}^{m}(F_{i})] = −Kδ^{\frac{1}{2}} Q_{ii,1}\{((q − q^{−1}) ∑_{k=1}^{r−1} Ċ_{ii}^{1−k}T_{ω_{i}}^{m+k}(F_{i})ψ̄_{i,r−k}^{(s)}) + Ċ_{ii}^{1−r}T_{ω_{i}}^{m+r}(F_{i})\},

[ψ̄_{ir}^{(s)}, T_{ω_{i}}^{m}(E_{i})] = K_{δ}^{− \frac{1}{2}} Q_{ii,1}\{((q − q^{−1}) ∑_{k=1}^{r−1} Ċ_{ii}^{k−1}T_{ω_{i}}^{m−k}(E_{i})ψ̄_{i,r−k}^{(s)}) + Ċ_{ii}^{r−1}T_{ω_{i}}^{m−r}(E_{i})\}.

(ii) Assume 1 ≤ i ≠ j ≤ n. Let r > 0 and m ∈ Z. Then:

[ψ̄_{ir}^{(s)}, T_{ω_{j}}^{m}(F_{j})]

= Kδ^{\frac{1}{2}} Q_{ij,1}\{((q − q^{−1}) ∑_{k=1}^{r−1}(−Ċ_{ij})^{1−k}T_{ω_{j}}^{m+k}(F_{j})ψ̄_{i,r−k}^{(s)}) + (−Ċ_{ij})^{1−r}T_{ω_{j}}^{m+r}(F_{j})\},

[ψ̄_{ir}^{(s)}, T_{ω_{j}}^{m}(E_{j})]

= −K_{δ}^{− \frac{1}{2}} Q_{ij,1}\{((q − q^{−1}) ∑_{k=1}^{r−1}(−Ċ_{ij})^{k−1}T_{ω_{j}}^{m−k}(E_{j})ψ̄_{i,r−k}^{(s)}) + (−Ċ_{ij})^{r−1}T_{ω_{j}}^{m−r}(E_{j})\}.

Let o(i) ∈ {±1} satisfy that o(i) ≠ o(j) if (α_{i}, α_{j}) ≠ 0 (i ≠ j). Put T̂_{ω_{i}}^{m}E_{i} = o(i)^{m}T_{ω_{i}}^{m}E_{i} and T̂_{ω_{i}}^{m}F_{i} = o(i)^{m}T_{ω_{i}}^{m}F_{i}. Define ˆ ψ̄_{ir}^{(s)} by replacing T_{ω_{i}} of (8.1.1) with T̂_{ω_{i}}. By Lemma 8.2.1, we have: Lemma 8.2.2. Assume j ≠ i or p(α_{i}) = 0. Then:

[ˆψ̄_{ir}^{(s)}, T̂_{ω_{j}}^{m}(F_{j})]

= −Kδ^{\frac{1}{2}} Q_{ij,1}\{((q − q^{−1}) ∑_{k=1}^{r−1} Ċ_{ij}^{1−k}T̂_{ω_{j}}^{m+k}(F_{j})ˆψ̄_{i,r−k}^{(s)}) + Ċ_{ij}^{1−r}T̂_{ω_{j}}^{m+r}(F_{j})\},

[ˆψ̄_{ir}^{(s)}, T̂_{ω_{j}}^{m}(E_{j})]

= K_{δ}^{− \frac{1}{2}} Q_{ij,1}\{((q − q^{−1}) ∑_{k=1}^{r−1} Ċ_{ij}^{k−1}T̂_{ω_{j}}^{m−k}(E_{j})ˆψ̄_{i,r−k}^{(s)}) + Ċ_{ij}^{r−1}T̂_{ω_{j}}^{m−r}(E_{j})\}.

Define h_{ik}^{(s)} ∈ U_{h}(G) (k > 0) by the following generating function in z.

exp((q − q^{−1}) ∑_{k=1}^{∞} h_{ik}^{(s)}z^{k}) = 1 + (q − q^{−1}) ∑_{k=1}^{∞} ˆ ψ̄_{ik}^{(s)}z^{k}.

Remark. For (α_{i}, α_{i}) = 0, we have not shown [ˆψ̄_{ik}^{(s)}, ˆ ψ̄_{ir}^{(s)}] = 0 yet. Hence an uncertainly of the definition of h_{ik}^{(s)} has still remined. It depends on an order of {ˆψ̄_{ik}^{(s)}}. By Lemma 8.2.2, we have: Lemma 8.2.3. Assume j ≠ i or (α_{i}, α_{i}) ≠ 0. Then:

\begin{matrix}[h_{ik}^{(s)}, T̂_{ω_{j}}^{m}(F_{j})] = −\frac{1}{k}Q_{ij,k}Kδ^{\frac{k}{2}} T̂_{ω_{j}}^{m+k}(F_{j}), \\ [h_{ik}^{(s)}, T̂_{ω_{j}}^{m}(E_{j})] = \frac{1}{k}Q_{ij,k}K_{δ}^{− \frac{k}{2}} T̂_{ω_{j}}^{m−k}(E_{j}). \end{matrix}

8.3. Lemma 8.3.1. Let 1 ≤ i ≤ n and r ∈ Z. Then:

\begin{matrix}[[T_{ω_{i}}^{m+r}(F_{i}), T_{ω_{i}}^{m}(F_{i})]] = −[[T_{ω_{i}}^{m+1}(F_{i}), T_{ω_{i}}^{m+r−1}(F_{i})]], \\ [[T_{ω_{i}}^{m}(E_{i}), T_{ω_{i}}^{m+r}(E_{i})]] = −[[T_{ω_{i}}^{m+r−1}(E_{i}), T_{ω_{i}}^{m+1}(E_{i})]]. \end{matrix}

Proof. For (α_{i}, α_{i}) ≠ 0, we have already known these by [B]. For (α_{i}, α_{i}) = 0, by Lemma 8.2.3 and T_{ω_{i}}^{m}(F_{i})^{2} = T_{ω_{i}}^{m}(E_{i})^{2} = 0,

[T_{ω_{i}}^{m+r}(F_{i}), T_{ω_{i}}^{m}(F_{i})] = [T_{ω_{i}}^{m+r}(E_{i}), T_{ω_{i}}^{m}(E_{i})] = 0,  (8.3.1)

which are nothing else but the formulae we want.

Q.E.D.

Lemma 8.3.2. Let (α_{i}, α_{i}) = 0 and r > 0. Then ψ̄_{ir}^{(s)} = ψ̄_{ir}^{(s′)} and

[h_{ir}^{(s)}, T_{ω_{i}}^{m}(F_{i})] = [h_{ir}^{(s)}, T_{ω_{i}}^{m}(E_{i})] = 0.  (8.3.2)

Proof. By (8.3.1), we have:

[E_{i}, ψ̄_{ir}^{(0)}] = [ψ̄_{ir}^{(0)}, F_{i}] = 0.  (8.3.3)

We use an induction on r. Let 1 ≤ j ≤ n be such that (α_{i}, α_{j}) ≠ 0. First we assume r = 1. Then h_{i1}^{(s)} = o(i)ψ̄_{i1}^{(s)}.

ψ̄_{i1}^{(−1)} = K_{δ}^{− \frac{1}{2}} [T_{ω_{i}}^{−1}(E_{i}), K_{i}^{−1}F_{i}]
= o(i)K_{δ}^{− \frac{1}{2}} Q_{ji,1}^{−1}Kδ^{\frac{1}{2}} [[h_{j1}^{(0)}, E_{i}], K_{i}^{−1}F_{i}] (by Lemma 8.2.3)
= o(i)Q_{ji,1}^{−1}K_{i}^{−1} · o(i)Q_{ji,1}[K_{δ}^{− \frac{1}{2}} T_{ω_{i}}(F_{i}), E_{i}]
= K_{δ}^{− \frac{1}{2}} [T_{ω_{i}}(K_{i}^{−1}F_{i}), E_{i}]
= ψ̄_{i1}^{(0)}.

Hence, by (8.3.3), we get our formulae for r = 1.

We assume that we have shown the lemma for 1, 2,..., r − 1. Firstly we show [h_{j1}^{(0)}, h_{jr−1}^{(0)}] = 0. By Lemma 8.2.3, we have:

[[h_{j1}^{(0)}, h_{jr−1}^{(0)}], T̂_{ω_{k}}^{m}(F_{k})] = [[h_{j1}^{(0)}, h_{jr−1}^{(0)}], T̂_{ω_{k}}^{m}(E_{k})] = 0

for 1 ≤ k ≤ n and m ∈ Z. By Lemma 8.2.1 (i), Kδ^{\frac{r}{2}} [h_{j1}^{(0)}, h_{jr−1}^{(0)}] ∈ N^{+}. We know the fact that T̂_{ω_{k}}^{m}(F_{k}), T̂_{ω_{k}}^{m}(E_{k}) and H generate U_{h}(G). Hence, by Proposition 6.2.1, we get

[h_{j1}^{(0)}, h_{jr−1}^{(0)}] = 0  (8.3.4)

as well as [h_{j1}^{(0)}, ψ̄_{jr−1}^{(0)}] = 0. Hence:

ψ̄_{ir}^{(−1)} = K_{δ}^{− \frac{1}{2}} [T_{ω_{i}}^{−1}(E_{i}), T_{ω_{i}}^{r−1}(K_{i}^{−1}F_{i})]
= o(i)K_{δ}^{− \frac{r}{2}} Q_{ji,1}^{−1}Kδ^{\frac{1}{2}} [[h_{j1}^{(0)}, E_{i}], T_{ω_{i}}^{r−1}(K_{i}^{−1}F_{i})] (by Lemma 8.2.3)
= o(i)Q_{ji,1}^{−1}Kδ^{\frac{1−r}{2}} \{[h_{j1}^{(0)}, Kδ^{\frac{r−1}{2}} ψ̄_{ir−1}^{(0)}] + o(i)Q_{ji,1}[K_{δ}^{− \frac{1}{2}} T_{ω_{i}}^{r}(K_{i}^{−1}F_{i}), E_{i}]\}
= ψ̄_{ir}^{(0)}.

Hence, by (8.3.3), we get our formulae.

Q.E.D.

We put h_{ir} = h_{ir}^{(s)} and ˆ ψ̄_{ik} ˆψ̄_{ik}^{(s)} (r > 0, 1 ≤ i ≤ n) which is well defined by Lemma 8.3.2. Similarly to show (8.3.4), we have: Lemma 8.3.3. [h_{ir}, h_{ir′}] = 0. By Lemma 8.2.3 and Lemma 8.3.2, we have: Lemma 8.3.4.

\begin{matrix}[h_{ik}, T̂_{ω_{j}}^{m}(F_{j})] = −\frac{1}{k}Q_{ij,k}Kδ^{\frac{k}{2}} T̂_{ω_{j}}^{m+k}(F_{j}), \\ [h_{ik}, T̂_{ω_{j}}^{m}(E_{j})] = \frac{1}{k}Q_{ij,k}K_{δ}^{− \frac{k}{2}} T̂_{ω_{j}}^{m−k}(E_{j}). \end{matrix}

We know that Kδ^{\frac{k}{2}} h_{ik} ∈ N^{+}. By Lemma 8.3.4 and Proposition 6.2.1, since T̂_{ω_{j}}^{m}(E_{j}), T̂_{ω_{j}}^{m}(F_{j}) and H generate U_{h}(G), we have: Lemma 8.3.5. Keep notations in 1.6. If ∑_{i=1}^{N} d̄_{i} = 0, then

∑_{i=1}^{n}[∑_{j=1}^{i} kd̄_{j}]Kδ^{\frac{k}{2}} h_{ik} = 0 (k ≥ 1).  (8.3.5)

in N^{+} ⊂ U_{h}(G)(E, Π, p) of (E, Π, p) of Diagram 1.6.2 (N ≥ 4). After all we have: Theorem 8.3.6. Let (E, Π, p) be the datum of Diagram 1.6.2 (N ≥ 4) with ∑ d̄_{i} = 0. Then the defining relations of U_{h}(G(E, Π, p)) are defined by adding (8.3.5) to the ones of Theorem 6.6.1. 8.4. For 1 ≤ i ≤ N − 1 and r ≥ 0, put

ψ_{ir} = \{ \begin{matrix}(q − q^{−1})K_{i} ˆψ̄_{ik} & (r > 0), \\ K_{i} & (r = 0). \end{matrix}

and φ_{ir} = Ω(ψ_{ir}). Put h_{i,−r} = h_{ir} (r > 0). For 1 ≤ i ≤ N − 1 and k ∈ Z, put x_{ik}^{−} = T̂_{ω_{i}}^{k}(F_{i}) and x_{ik}^{+} = T̂_{ω_{i}}^{−k}(E_{i}). Similar to [B], we have: Theorem 8.4.1. Let (E, Π, p) be the datum of Diagram 1.6.2 (N ≥ 3). Then U_{h}(G(E, Π, p)) is defined with the generators {H ∈ H, x_{ij}^{±}, h_{ik}} and the relations:

[H, x_{jk}^{±}] = (±α_{j} + kδ)(H)x_{jk}^{±},

[h_{ik}, h_{jl}] = δ_{k,−l} \frac{1}{k}Q_{ij,k} \frac{K_{δ}^{k}−K_{δ}^{−k}}{q−q^{−1}} ,

x_{ik+1}^{±}x_{jl}^{±} − (−1)^{p(α_{i})p(α_{j})}q^{±(α_{i},α_{j})}x_{jl}^{±}x_{ik+1}^{±} = (−1)^{p(α_{i})p(α_{j})}q^{±(α_{i},α_{j})}x_{ik}^{±}x_{jl+1}^{±} − x_{jl+1}^{±}x_{ik}^{±},

[x_{ik}^{+}, x_{jl}^{−}] = δ_{ij} ^{\frac{K \frac{k−l}{2} ; δ ψ_{ik+l}−K \frac{l−k}{2} ; δ ϕ_{ik+l}}{q−q^{−1}}} ,

[x_{ik}^{±}, x_{il}^{±}] = 0 if (α_{i}, α_{i}) = 0,

(In the following equations, Sym_{k_{1},k_{2},...,k_{s}} means symmetrization with respect to {k_{1}, k_{2}, …, k_{s}}.)

Sym_{k_{1},k_{2}}[[x_{ik_{1}}^{±}, [[x_{ik_{2}}^{±}, x_{jl}^{±}]]]] = 0 if (α_{i}, α_{i}) ≠ 0 and (α_{i}, α_{j}) ≠ 0,

Sym_{k_{1},k_{2}}[[[[[x_{il}^{±}, x_{jk_{1}}^{±}]], x_{um}^{±}]], x_{jk_{2}}^{±}] = 0  if ^{i}  ^{j}  ^{u} × ^{}× ×

(Each of the following equations means an equation as a generate function in an indeterminate z.)

\begin{matrix}∑_{k≥0} ψ_{ik}z^{k} = K_{i} exp((q − q^{−1}) ∑_{r≥1} h_{ir}z^{r}), \\ ∑_{k≥0} ϕ_{ik}z^{k} = K_{i}^{−1} exp((q^{−1} − q) ∑_{r≥1} h_{i,−r}z^{r}). \end{matrix}

Acknowledgment. I would like to thank Professor T.Tanisaki for teaching me about Lemma 6.2.2.

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