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, instead of geometric, method of affine quantum Schur-Weyl conception. to start, a variety of algebraic constructions are mentioned, together with double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. the remainder of the publication investigates the affine quantum Schur-Weyl duality on 3 degrees. This comprises 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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Schiffmann–Hubery generators 37 where I denotes the ideal generated by 1 ⊗ K α − K α ⊗ 1, for all α ∈ ZI . By the construction, I is indeed a Hopf ideal of D (n). Thus, D (n) is again a Hopf algebra. We call D (n) the double Ringel–Hall algebra of the cyclic quiver (n). , D (n)− ) be the Q(v)-subalgebra of D (n) generated − + by u + (n). , u A ) for all A ∈ D (n) generated by K α for all α ∈ ZI . Then D (n)+ ∼ = H (n), D (n)− ∼ = H (n)op , and D (n)0 ∼ = Q(v)[K 1±1 , . . , K n±1 ]. 3) Moreover, the multiplication map D (n)+ ⊗ D (n)0 ⊗ D (n)− −→ D (n) is an isomorphism of Q(v)-vector spaces.

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 , .

5. The double Ringel–Hall algebra D (n) is a Hopf algebra with comultiplication , counit ε, and antipode σ defined by (E i ) = E i ⊗ K i + 1 ⊗ E i , (K i±1 ) = K i±1 ⊗ K i±1 , (Fi ) = Fi ⊗ 1 + K i−1 ⊗ Fi , ± ± (z± s ) = zs ⊗ 1 + 1 ⊗ zs ; ε(E i ) = ε(Fi ) = 0 = ε(z± s ), σ (E i ) = −E i K i−1 , ε(K i ) = 1; σ (Fi ) = − K i Fi , σ (K i±1 ) = K i∓1 , ± and σ (z± s ) = −zs , where i ∈ I and s ∈ J∞ . 6. (1) For notational simplicity, we sometimes continue to use u i± as generators of D (n). 4. Some integral forms for i ∈ I and s 45 1.

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

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