By Bangming Deng

ISBN-10: 1607092050

ISBN-13: 9781607092056

The speculation of Schur-Weyl duality has had a profound impression over many components of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and offers an algebraic, rather than geometric, method of affine quantum Schur-Weyl idea. 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 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 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.

**Read or Download A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory PDF**

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

**Sample text**

2) (2) (2) = (b3 ∗ a2 )a2 ψ(a1 , b2 )ψ(a1 , b1 ) = a1 ∗ (b2 ∗ a2 )ψ(a2 , b3 )ψ(a1 , b1 ) (by induction) = (2) (2) (2) a1 a1 ∗ b1 ψ(a2 , b3 )ψ(a2 , b2 ) (since a ∈ X A ) = a1 a1 ∗ b1 ψ(a2 a2 , b2 ), (2) (2) B (b) = (2) (2) b1(2) ⊗ b2(2) ⊗ b3(2) . Hence, h a,b ∈ I . 3). The following lemma can be checked directly. 2. Let A = (A , μA , ηA , A , εA , σA ), B = (B, μB , ηB , B , εB , σB ) be two Hopf algebras over a field F together with a skewHopf pairing ψ : A × B → F. Assume that σA and σB are both invertible.

2. Let A = (A , μA , ηA , A , εA , σA ), B = (B, μB , ηB , B , εB , σB ) be two Hopf algebras over a field F together with a skewHopf pairing ψ : A × B → F. Assume that σA and σB are both invertible. , op op −1 −1 (A , μA , ηA , A , εA , σA ) and (B, μB , ηB , B , εB , σB )). 1. 5. First, we need a skew-Hopf pairing. 2 to [78, Prop. 3] yields the following result. For completeness, we sketch a proof. We introduce some notation which is used in the proof. For each α = i∈I ai i ∈ + ZI , write τ α = (n), we have i ∈I ai−1 i .

5 to obtain double Ringel–Hall algebras D (n) of cyclic quivers. The algebras D (n) possess a rich structure. 1). 3). 3). 4. 1. Drinfeld doubles and the Hopf algebra D (n) In this section, we first recall from [46] the notion of a skew-Hopf pairing and define the associated Drinfeld double. 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 Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory by Bangming Deng

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