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

By Bangming Deng

Similar algebra & trigonometry books

Approximation Theorems of Mathematical Statistics (Wiley Series in Probability and Statistics)

This paperback reprint of 1 of the simplest within the box covers a vast diversity of restrict theorems worthwhile in mathematical records, besides equipment of facts and strategies of program. The manipulation of "probability" theorems to procure "statistical" theorems is emphasised.

4000 Jahre Algebra: Geschichte – Kulturen – Menschen

Die Entstehung, Entwicklung und Wandlung der Algebra als Teil unserer Kulturgeschichte beschreiben Wissenschaftler von fünf Universitäten. Ursprünge, Anstöße und die Entwicklung algebraischer Begriffe und Methoden werden in enger Verflechtung mit historischen Ereignissen und menschlichen Schicksalen dargestellt.

Rings, Extensions, and Cohomology

"Presenting the complaints of a convention held lately at Northwestern college, Evanston, Illinois, at the social gathering of the retirement of famous mathematician Daniel Zelinsky, this novel reference presents updated insurance of themes in commutative and noncommutative ring extensions, particularly these regarding problems with separability, Galois thought, and cohomology.

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

Sample text

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!