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

By Bangming Deng

ISBN-10: 1607092050

ISBN-13: 9781607092056

The speculation of Schur-Weyl duality has had a profound effect over many components of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and offers an algebraic, instead of geometric, method of affine quantum Schur-Weyl idea. to start, quite a few algebraic constructions are mentioned, together with double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. the remainder of the ebook investigates the affine quantum Schur-Weyl duality on 3 degrees. This contains the affine quantum Schur-Weyl reciprocity, the bridging function of affine q-Schur algebras among representations of the quantum loop algebras and people of the corresponding affine Hecke algebras, presentation of affine quantum Schur algebras and the realisation conjecture for the double Ringel-Hall algebra with an explanation of the classical case. this article is perfect for researchers in algebra and graduate scholars who are looking to grasp Ringel-Hall algebras and Schur-Weyl duality.

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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 , .

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A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory by Bangming Deng


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