By Bangming Deng
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
Similar algebra & trigonometry books
A glance at baseball facts from a statistical modeling point of view! there's a fascination between baseball enthusiasts and the media to gather facts on each possible occasion in the course of a 3-hitter and this e-book addresses a couple of questions which are of curiosity to many baseball enthusiasts. those contain tips to fee avid gamers, expect the end result of a online game or the attainment of an success, making feel of situational info, and finding out the main important gamers on this planet sequence.
This ebook is a self-contained straightforward creation to jewelry and Modules, a subject constituting approximately half a center path on Algebra. The proofs are handled with complete info protecting the study room flavour. the whole fabric together with workout is totally classification demonstrated. True/False statements are intended for a fast attempt of figuring out of the most textual content.
The paintings of facts is designed for a one-semester or two-quarter direction. a standard pupil may have studied calculus (perhaps additionally linear algebra) with average good fortune. With an crafty mix of chatty kind and engaging examples, the student's earlier intuitive wisdom is put on sturdy highbrow floor.
Combinatorial layout idea is a resource of easily said, concrete, but tricky discrete difficulties, with the Hadamard conjecture being a major instance. It has turn into transparent that lots of those difficulties are primarily algebraic in nature. This publication offers a unified imaginative and prescient of the algebraic topics that have built up to now in layout conception.
- 100 Algebra Workouts
- Essentials of College Physics ,
- Recent Progress in Algebra: An International Conference on Recent Progress in Algebra, August 11-15, 1997, Kaist, Taejon, South Korea
- Degree Theory for Equivariant Maps, the General S1-Action
Extra info for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory
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  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  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