The quantum double <math> <mi>D</mi><mfenced> <mi>A</mi> </mfenced> </math>

The quantum double D A

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 10 April 2010

The quantum double D A

In general it can be very difficult to find quasitriangular Hopf algebras, especially ones where the element is different from 11. The construction in (???) belows says that, given a Hopf algebra A we can sort of paste it and its dual A* together to get a quasitriangular Hopf algebra D A and that the for this new quasitriangular Hopf algebra is both a natural one and is nontrivial.

Let A= AmΔiεS be a Hopf algebra over 𝕂. Let A*= Hom 𝕂 A𝕂 be the dual of A. There is a natural bilinear pairing , :A*A𝕂 between A and A* given by αa =α a ,for allαA*  and  aA. Extend this notation so that if α1 , α 2 A* and a 1 , a 2 A then α 1 α 2 a 1 a 2 = α1 a 1 α 2 a 2 . We make A* into a Hopf algebra, which is denoted A *coop , by defining a multiplication and a comultiplication Δ on A* via the equations α1 α 2 a = α1 α 2 Δ a and Δ op α a 1 a 2 = α a1 a 2 , for all α, α1 α2 A* and a, a 1 , a 2 A. The definition of Δ op is in (4.1).

  1. The identity in A *coop is the counit ε:A𝕂.
  2. The counit of A *coop is the map ε: A* 𝕂 α α 1 where  1  is the identity in  A.
  3. The antipode of A *coop is given by teh identity S α ia = α S -1 a , for all αA* and all aA.

We want to paste the algebras A and A *coop together in order to make a quasi triangular Hopf algebra by D A . There are three main steps:

  1. We paste the two together by letting D A =A A *coop . Write elements of D A as aα instead of aα.
  2. We want the multiplication in D A to reflect the multiplication in A and the multiplication in A *coop . Similarly for the comultiplication.
  3. We want the -matrix to be = i b i bi , where b i is a basis of A and b i is the dual basis in A*.
The condition in (2) determines the comultiplication in D A , Δ aα =Δ a Δ α = a,α a 1 α 1 a 2 α 2 , where Δ a = a a 1 a 2 and Δ α = α α 1 α 2 . The condition in (2) doesn't quite determine the multiplication in D A . We need to be able to expand products like a 1 α 1 a 2 α 2 . If we knew α 1 a 2 = j b j β j ,for some β j A *coop   and   b j A, then we would have a 1 α 1 a 2 α 2 = j a 1 bj β j α2 which is a well defined element of D A . Miraculously, the condition in (3) and the equation Δ a -1 = Δ op a ,for allaA, force that if α A *coop and aA then, in D A , αa= α,a α 1 S -1 a 1 α 3 a 3 a 2 α 2 , and αa= α,a α 1 a 1 α 3 S -3 a 1 α 2 a 2 , where, if Δ is the comultiplication in D A , Δid Δ a = a a 1 a 2 a 3 ,and Δid Δ α = α α 1 α 2 α 3 .

These relations completely determine the multiplication in D A . This construction is summarised in the following theorem.

Let A be a (finite dimensional) Hopf algebra over 𝕂 and let A *coop be the Hopf algebra A*= Hom 𝒦 A𝕂 except with opposite comultiplication. Then there exists a unique quasitriangular Hopf algebra D A given by

  1. The 𝕂 -linear map AA* D A aα aα is bijective.
  2. D A contains A and A *coop as Hopf subalgebras.
  3. The element D A D A is given by = i b i b i , where b i is a basis of A and b i is the dual basis in A *coop .

In condition (2) of the theorem, A is identified withthe image of A1 under the map in (1) and A *coop is identified with the image of 1 A *coop under the map in (1).

The following proposition constructs an ad-invariant bilinear form on D A .

Let A be a Hopf algebra. The bilinear form on the quantum double of D(A) of A which is defined by aα bβ = β S a α S -1 b ,for alla,bA  and all  α,β A *coop , satisfies ad a x y = x ad S u y and yx = x S 2 y , for all u,x and yD A .

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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