By Bangming Deng

ISBN-10: 1607092050

ISBN-13: 9781607092056

The idea of Schur-Weyl duality has had a profound impact over many components of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and provides an algebraic, in place of geometric, method of affine quantum Schur-Weyl thought. to start, quite a few algebraic buildings are mentioned, together with double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. the remainder of the booklet investigates the affine quantum Schur-Weyl duality on 3 degrees. This contains the affine quantum Schur-Weyl reciprocity, the bridging position 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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**Additional info for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory**

**Example text**

If we denote the standard basis of X by e1 , . . , en and the dual basis by f 1 , . . , f n , then f i , e j rd = δi, j . 1) with en+1 = e1 and f n+1 = f 1 define a root datum (Y, X, , rd , . ). For notational simplicity, we shall identify both X and Y with ZI by setting ei = i = f i for all i ∈ I . Under this identification, the form , rd : ZI × ZI → Z becomes a symmetric bilinear form, which is different from the Euler forms , and its symmetrization ( , ). However, they are related as follows.

More precisely, we will modify the construction in [40] to extend Beck’s algebra embedding of the quantum affine sln into the quantum loop algebra U(gln ) defined by Drinfeld’s new presentation [20] to an explicit isomorphism from D (n) to U(gln ). 1, consider the loop algebra gln = gln ⊗ Q[t, t −1 ], 50 2. Double Ringel–Hall algebras of cyclic quivers which is generated by E i,i+1 ⊗ t s , E i +1,i ⊗ t s , and E j, j ⊗ t s , for s ∈ Z, 1 i n − 1, and 1 j n. We have the following quantum enveloping algebra associated with gln ; see [20] (and also [28, 40]).

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. Some integral forms 47 that V 0 is Z-free. 14] (see also [22] and [12, Th. 20]) shows that V 0 contains the Z-subalgebra V 1 generated by K i±1 and K it ;0 which has a Z-basis n a K= Ki i i=1 n−1 a K = Ki i i=1 Ki ; 0 ti ai ∈ {0, 1}, ti ∈ N, ∀i ∈ I , Ki ; 0 Kn ; 0 · K ian ti tn ai ∈ {0, 1}, ti ∈ N, ∀i ∈ I .

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

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