# Read e-book online A Double Hall Algebra Approach to Affine Quantum Schur-Weyl PDF

By Bangming Deng

ISBN-10: 1607092050

ISBN-13: 9781607092056

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Extra resources for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory

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Fi /[m] , where i ∈ I , t 0, and s, m 1. Further, we set U + = U+ ∩ U , U − = U− ∩ U , and U 0 = U0 ∩ U . , ys and Fi ), for i ∈ I and s, m 1. Consequently, we obtain U + = U (sln )+ ⊗ Z[x1 , x2 , . ] and U − = U (sln )− ⊗ Z[y1 , y2 , . ], where U (sln )+ and U (sln )− are the Z-subalgebras of U(sln ) generated by (m) (m) the divided powers E i and Fi , respectively. This implies in particular that both U + and U − are free Z-modules. We now look at the structure of U 0 . Let V 0 be the Z-subalgebra generated ks ;0 by K i±1 , K it ;0 , k±1 0, and s 1.

4 induces a surjective Z-algebra homomorphism : U → D (n). We also set D (n) = D (n) ∩ D (n) for ∈ {+, −, 0}. , U − ). Hence, + D (n)+ = C (n)+ ⊗Z Z[z+ 1 , z2 , . ] and − D (n)− = C (n)− ⊗Z Z[z− 1 , z2 , . , (u i− )(m) ). 2). 1) is a Z-module isomorphism; see [12, Cor. 50]. In particular, D (n) is a free Z-module. The Z-algebra D (n) gives rise to a Z-form U (n) for U (n), the extended quantum affine sln : U (n) = D (n) ∩ U (n) = C (n)+ D (n)0 C (n)− . 4 in terms of specialization. ks ;0 3 Of course, this fact can be proved directly by the relation [x , y ] = δ s t s,t 1 .

0 0 0 · · · v ±(n−2)s −v ±ns ⎠ 1 1 1 1 1 ∓ [s] ∓ [s] ∓ [s] ··· ∓ [s] ∓ [s] By the definition of hi,±s and θ±s , n hi,±s = n (±s) X i, j g j,±s , for 1 i < n, and θ±s = j =1 (±s) X n, j g j,±s . j =1 A direct calculation shows that det(X (±s) ) = ∓ 1 1 + v ±2s + · · · + v ±2(n−1)s [s] n−2 v ±is = 0 (n (±s) We denote the inverse of X (±s) by Y (±s) = (Yi, j ). Thus, for each 1 n−1 gi,±s = (±s) 2). i=1 i n, (±s) Yi, j h j,±s + Yi,n θ±s . j =1 Therefore, the Q(v)-subspace of U(gln ) spanned by g1,±s , .