By Bangming Deng

ISBN-10: 1607092050

ISBN-13: 9781607092056

The idea of Schur-Weyl duality has had a profound impact over many parts 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 concept. 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 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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**Extra info for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory**

**Example text**

Let C (n) = C (n) ⊗Z Q(v). Thus, C (n) identifies with the Q(v)-subalgebra H (n) generated by u i = u [Si ] for i ∈ I . 4. ([65]) There are Q(v)-algebra isomorphisms C (n) −→ U(sln )+ , u i −→ E i and C (n)op −→ U(sln )− , u i −→ Fi . 2 gives rise to a monomial basis for U(sln ). 4. Three types of generators and associated monomial bases In this section, we display three distinct minimal sets of generators for H (n), each of which contains the generators {u i }i∈I for C (n). We also describe their associated monomial bases for H (n) in the respective generators.

Proof. The Hopf structure on H (n) 0 is almost identical to the Hopf algebra H defined in the proof of [78, Prop. 8] except that we used K α instead of K α in the comultiplication and antipode. Thus, the comultiplication of H (n) 0 defined here is opposite to that defined in [78, Th. 5], while the antipode is the inverse. 3, H (n) 0 is the Hopf algebra semi-opposite to a variant of the Hopf algebras considered in [78, loc cit]. (Of course, one can directly check by mimicking the proof of [78, Th.

The rest of the proof is clear. We now look at an integral form of D (n). 4. The integral form D (n) of D (n) is the Z-subalgebra gen+ (m) − (m) , for i ∈ I and s, t, erated by K i±1 , K it ;0 , z+ and (u − s , zs , (u i ) i ) m 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.

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

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