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

By Bangming Deng

The idea 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, in place of geometric, method of affine quantum Schur-Weyl concept. to start, numerous algebraic constructions are mentioned, together with double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. the remainder of the e-book 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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**Additional info for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory**

**Example text**

Let m ∈ Z+ ∪{∞} and Cm be defined as above. 3. Presentation of D (n) 43 (R7) E i xs = xs E i , xs xt = xt xs ; (R8) Fi ys = ys Fi , ys yt = yt ys . Moreover, U(Cm ) is a Hopf algebra with comultiplication antipode σ defined by (E i ) = E i ⊗ K i + 1 ⊗ E i , (xs ) = xs ⊗ ks + 1 ⊗ xs , (K i±1 ) = K i±1 ⊗ K i±1 , (Fi ) = Fi ⊗ 1 + K i−1 ⊗ Fi , (ys ) = ys ⊗ 1 + k−1 s ⊗ ys , ±1 ±1 (k±1 s ) = ks ⊗ ks ; ε(E i ) = ε(xs ) = 0 = ε(Fi ) = ε(ys ), σ (E i ) = σ (xs ) = −E i K i−1 , −xs k−1 s , , counit ε, and σ (Fi ) = − K i Fi , ε(K i ) = ε(ks ) = 1; σ (K i±1 ) = K i∓1 , ∓1 σ (ys ) = −ks ys , and σ (k±1 s ) = ks , where i ∈ I and s ∈ Jm .

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 . In particular, for each A ∈ τ d(A) = d(τ (A)). Then, for α, β ∈ ZI , α, β = (α − τ α) β = − β, τ α and K α = K α−τ α = K 1a1 · · · K nan . 3. 1) (n), is a skew-Hopf pairing. Proof. Condition (HP1) is obvious. We now check condition (HP2). Without − − loss of generality, we take a = u + A K α , b = K β u B , and b = K γ u C for + α, β, γ ∈ ZI and A, B, C ∈ (n).

Therefore, φ CA = 0. 1) that φ EB m B ∈ H (n)(m−1) , E u E − φC E u mδ = φ E m E + B