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

By Bangming Deng

The idea of Schur-Weyl duality has had a profound effect over many parts of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and provides an algebraic, in preference to geometric, method of affine quantum Schur-Weyl thought. to start, a variety of 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 evidence 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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Extra resources for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory

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1]. 3. If A = (A , μ, η, , ε, σ ) is a Hopf algebra with multiplication μ, unit η, comultiplication , counit ε, and antipode σ , then A op = (A , μop , η, op , ε, σ ) is also a Hopf algebra. This is called the opposite Hopf algebra of A . Moreover, if σ is invertible, then both (A , μop , η, , ε, σ −1 ) and (A , μ, η, op , ε, σ −1 ) are also Hopf algebras, which are called semiopposite Hopf algebras. 5. Hopf structure on extended Ringel–Hall algebras 27 Let H (n) 0 = H (n) ⊗Q(v) Q(v)[K 1±1 , . .

We then apply this general construction to obtain the Drinfeld double D (n) of the Ringel–Hall algebra H (n); see [78] for a general construction. Let A = (A , μA , ηA , A , εA , σA ) and B = (B, μB , ηB , B , εB , σB ) be two Hopf algebras over a field F. A skew-Hopf pairing of A and B is an F-bilinear form ψ : A × B → F satisfying: (HP1) ψ(1, b) = εB (b), ψ(a, 1) = εA (a), for all a ∈ A , b ∈ B; (HP2) ψ(a, bb ) = ψ( A (a), b ⊗ b ), for all a ∈ A , b, b ∈ B; 31 32 2. Double Ringel–Hall algebras of cyclic quivers op (HP3) ψ(aa , b) = ψ(a ⊗ a , B (b)), for all a, a ∈ A , b ∈ B; −1 (HP4) ψ(σA (a), b) = ψ(a, σB (b)), for all a ∈ A , b ∈ B, op op where ψ(a ⊗ a , b ⊗ b ) = ψ(a, b)ψ(a , b ), and B is defined by B (b) = b2 ⊗ b1 if B (b) = b1 ⊗ b2 .

For m, r, s min{r,s} [xrm , ysm ] = i =1 r i 1, s km − k−1 m i! i v − v −1 i r−i ys−i m xm . 1) 46 2. Double Ringel–Hall algebras of cyclic quivers Proof. Since [xm , ym ] = direct to check that, for s km −k−1 m v−v −1 km ;0 1 = ± and [xm , k± m ] = 0 = [ym , km ], it is 1, km − k−1 m s−1 y . 1). The case r = 1 is given as above. 1) holds for r . If r < s, then [xm , ysm ] = s [xrm+1 , ysm ] = xm [xrm , ysm ] + [xm , ysm ]xrm r = i=1 r = i=1 r i s km − k−1 m i! i v − v −1 i r i s km − k−1 m i! i v − v −1 i +s r+1 i=2 s km − k−1 m i!

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