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

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 Extend this notation so that if ${\alpha }_{1},{\alpha }_{2}\in A*$ and ${a}_{1},{a}_{2}\in A$ 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 ${A}^{*\mathrm{coop}},$ by defining a multiplication and a comultiplication $\Delta$ 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 $\alpha ,{\alpha }_{1}{\alpha }_{2}\in A*$ 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
3. The antipode of ${A}^{*\mathrm{coop}}$ is given by teh identity $⟨S\left(\alpha \right)ia⟩=⟨\alpha ,{S}^{-1}\left(a\right)⟩,$ 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 $D A =A⊗ A *coop .$ 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 $ℛ= ∑ i b i ⊗ bi ,$ 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 $D\left(A\right),$ $Δ aα =Δ a Δ α = ∑ a,α a 1 α 1 ⊗ a 2 α 2 ,$ 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 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 $D\left(A\right).$ Miraculously, the condition in (3) and the equation $ℛΔ a ℛ -1 = Δ op a ,for alla∈A,$ force that if $\alpha \in {A}^{*\mathrm{coop}}$ and $a\in A$ then, in $D\left(A\right),$ $αa= ∑ α,a α 1 S -1 a 1 α 3 a 3 a 2 α 2 ,$ and $αa= ∑ α,a α 1 a 1 α 3 S -3 a 1 α 2 a 2 ,$ where, if $\Delta$ is the comultiplication in $D\left(A\right),$$Δ⊗id Δ a = ∑ a a 1 ⊗ a 2 ⊗ a 3 ,and Δ⊗id Δ α = ∑ α α 1 ⊗ α 2 ⊗ α 3 .$

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 $A⊗A* → D A a⊗α ↦ aα$ 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 $ℛ= ∑ i b i ⊗ b i ,$ 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 satisfies $ad a x y = x ad S u y$and yx = x S 2 y , for all $u,x$ and $y\in D\left(A\right).$