The quantum double $D\left(A\right)$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 April 2010

The quantum double $D\left(A\right)$

In general it can be very difficult to find quasitriangular Hopf algebras, especially ones where the element $ℛ$ is different from $1\otimes 1.$ 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 $D\left(A\right)$ and that the $ℛ$ for this new quasitriangular Hopf algebra is both a natural one and is nontrivial.

Let $A=\left(A,m,\Delta ,i,\epsilon ,S\right)$ be a Hopf algebra over $𝕂.$ Let $A*={\mathrm{Hom}}_{𝕂}\left(A,𝕂\right)$ be the dual of $A.$ There is a natural bilinear pairing $⟨,⟩:A*\otimes A\to 𝕂$ between $A$ and $A*$ given by $$⟨\alpha ,a⟩=\alpha \left(a\right),\text{for all}\alpha \in A*\text{ and }a\in A.$$ Extend this notation so that if ${\alpha }_1,{\alpha }_2\in A*$ and $a_1,a_2\in A$ then $$⟨{\alpha }_1\otimes {\alpha }_2,a_1\otimes a_2⟩=⟨{\alpha }_1,a_1⟩⟨{\alpha }_2,a_2⟩.$$ We make $A*$ into a Hopf algebra, which is denoted $A^{*\mathrm{coop}},$ by defining a multiplication and a comultiplication $\Delta$ on $A*$ via the equations $$⟨{\alpha }_1{\alpha }_2,a⟩=⟨{\alpha }_1\otimes {\alpha }_2,\Delta \left(a\right)⟩\text{and}⟨{\Delta }^{\mathrm{op}}\left(\alpha \right),a_1\otimes a_2⟩=⟨\alpha ,a_1{a}_2⟩,$$ for all $\alpha ,{\alpha }_1{\alpha }_2\in A*$ and $a,a_1,a_2\in A.$ The definition of ${\Delta }^{\mathrm{op}}$ is in (4.1).

1. The identity in $A^{*\mathrm{coop}}$ is the counit $\epsilon :A\to 𝕂.$
2. The counit of $A^{*\mathrm{coop}}$ is the map $$\begin{array}{rcll}\epsilon :& A*& \to & 𝕂\\ & \alpha & \mapsto & \alpha \left(1\right)\end{array}\text{where }1\text{ is the identity in }A.$$
3. The antipode of $A^{*\mathrm{coop}}$ is given by teh identity $⟨S\left(\alpha \right)⟩=⟨\alpha ,S^{-1}\left(a\right)⟩,$ for all $\alpha \in A*$ and all $a\in A.$

We want to paste the algebras $A$ and $A^{*\mathrm{coop}}$ together in order to make a quasi triangular Hopf algebra by $D\left(A\right)$. There are three main steps:

1. We paste the two together by letting $$D\left(A\right)=A\otimes A^{*\mathrm{coop}}.$$ Write elements of $D\left(A\right)$ as $a\alpha$ instead of $a\otimes \alpha .$
2. We want the multiplication in $D\left(A\right)$ to reflect the multiplication in $A$ and the multiplication in $A^{*\mathrm{coop}}.$ Similarly for the comultiplication.
3. We want the $ℛ$-matrix to be $$ℛ=\sum _i{b}_i\otimes b^i,$$ where $\left\{b_i\right\}$ is a basis of $A$ and $\left\{b^i\right\}$ is the dual basis in $A*.$

The condition in (2) determines the comultiplication in $D\left(A\right),$ $$\Delta \left(a\alpha \right)=\Delta \left(a\right)\Delta \left(\alpha \right)=\sum _{a,\alpha }{a}_{\left(1\right)}{\alpha }_{\left(1\right)}\otimes a_{\left(2\right)}{\alpha }_{\left(2\right)},$$ where $\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}$ and $\Delta \left(\alpha \right)=\sum _{\alpha }{\alpha }_{\left(1\right)}\otimes {\alpha }_{\left(2\right)}.$ The condition in (2) doesn't quite determine the multiplication in $D\left(A\right).$ We need to be able to expand products like $\left(a_1{\alpha }_1\right)\left(a_2{\alpha }_2\right).$ If we knew $${\alpha }_1{a}_2=\sum _j{b}_j{\beta }_j,\text{for some}{\beta }_j\in A^{*\mathrm{coop}}\text{ and }{b}_j\in A,$$ then we would have $$\left(a_1{\alpha }_1\right)\left(a_2{\alpha }_2\right)=\sum _j\left(a_1{b}_j\right)\left({\beta }_j{\alpha }_2\right)$$ which is a well defined element of $D\left(A\right).$ Miraculously, the condition in (3) and the equation $$ℛ\Delta \left(a\right){ℛ}^{-1}={\Delta }^{\mathrm{op}}\left(a\right),\text{for all}a\in A,$$ force that if $\alpha \in A^{*\mathrm{coop}}$ and $a\in A$ then, in $D\left(A\right),$ $$\alpha a=\sum _{\alpha ,a}⟨{\alpha }_{\left(1\right)},S^{-1}\left(a_{\left(1\right)}\right)⟩⟨{\alpha }_{\left(3\right)},a_{\left(3\right)}⟩a_{\left(2\right)}{\alpha }_{\left(2\right)},$$ and $$\alpha a=\sum _{\alpha ,a}⟨{\alpha }_{\left(1\right)},a_{\left(1\right)}⟩⟨{\alpha }_{\left(3\right)},S^{-3}\left(a_{\left(1\right)}\right)⟩{\alpha }_{\left(2\right)}{a}_{\left(2\right)},$$ where, if $\Delta$ is the comultiplication in $D\left(A\right),$$$\left(\Delta \otimes \mathrm{id}\right)\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}\otimes a_{\left(3\right)},\text{and}\left(\Delta \otimes \mathrm{id}\right)\Delta \left(\alpha \right)=\sum _{\alpha }{\alpha }_{\left(1\right)}\otimes {\alpha }_{\left(2\right)}\otimes {\alpha }_{\left(3\right)}.$$

These relations completely determine the multiplication in $D\left(A\right).$ This construction is summarised in the following theorem.

Let $A$ be a (finite dimensional) Hopf algebra over $𝕂$ and let $A^{*\mathrm{coop}}$ be the Hopf algebra $A*={\mathrm{Hom}}_{𝒦}\left(A,𝕂\right)$ except with opposite comultiplication. Then there exists a unique quasitriangular Hopf algebra $\left(D\left(A\right),ℛ\right)$ given by

1. The $𝕂$-linear map $$\begin{array}{rcl}A\otimes A*& \to & D\left(A\right)\\ a\otimes \alpha & \mapsto & a\alpha \end{array}$$ is bijective.
2. $D\left(A\right)$ contains $A$ and $A^{*\mathrm{coop}}$ as Hopf subalgebras.
3. The element $ℛ\in D\left(A\right)\otimes D\left(A\right)$ is given by $$ℛ=\sum _i{b}_i\otimes b^i,$$ where $\left\{b_i\right\}$ is a basis of $A$ and $\left\{b^i\right\}$ is the dual basis in $A^{*\mathrm{coop}}.$

In condition (2) of the theorem, $A$ is identified withthe image of $A\otimes 1$ under the map in (1) and $A^{*\mathrm{coop}}$ is identified with the image of $1\otimes A^{*\mathrm{coop}}$ under the map in (1).

The following proposition constructs an ad-invariant bilinear form on $D\left(A\right).$

Let A be a Hopf algebra. The bilinear form on the quantum double of D(A) of A which is defined by $$⟨a\alpha ,b\beta ⟩=⟨\beta ,S\left(a\right)⟩⟨\alpha ,S^{-1}\left(b\right)⟩,\text{for all}a,b\in A\text{ and all }\alpha ,\beta \in A^{*\mathrm{coop}},$$ satisfies $$⟨{\mathrm{ad}}_a\left(x\right),y⟩=⟨x,{\mathrm{ad}}_{S\left(u\right)}\left(y\right)⟩$$and yx = x S 2 y , for all $u,x$ and $y\in D\left(A\right).$

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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