I. Hopf algebras and quasitriangular Hopf algebras

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

Last update: 28 October 2012

Let $k$ be a field. Unless otherwise specified all maps between vector spaces over $k$ are assumed to be $k\text{-linear}$ and, if $V$ is a vector space over $k,$ then ${\text{id}}_V:V\to V$ denotes the identity map from $V$ to $V\text{.}$

The proofs of most of the statements in this chapter can be found in [Mon1992]. The proof that the antipode is an antihomomorphism (2.1) is given in [Swe1969] 4.0.1. The statement of Theorem (5.3), giving the construction of the quantum double, is given explicitly in [Dri1987] §13, and the proof can be found in [Maj1995] p. 2870289. A statement similar to Proposition (5.5) is in [Tan1992] Prop. 2.2.1 and the proof is similar to the proof given here.

SRMCwMFFs

Definition of an algebra

An algebra over $k$ is a vector space $A$ over $k$ with a multiplication

$$\begin{array}{cccc}m:& A\otimes A& ⟶& A\\ & a\otimes b& ⟼& a·b=ab\end{array}$$

and an identity element $1_A\in A$ such that

1. $m$ is associative, i.e. $\left(ab\right)c=a\left(bc\right),$ for all $a,b,c\in A,$ and
2. $1_A·a=a·1_A=a,$ for all $a\in A\text{.}$

Equivalently, an algebra over $k$ is a vector space $A$ over $k$ with a multiplication $m:A\otimes A\to A$ and a unit $\iota :k\to A$ such that

1. $m$ is associative, i.e. $m\circ (m\otimes {\text{id}}_A)=m\circ ({\text{id}}_A\otimes m),$ and
2. (unit condition) $m\circ (\iota \otimes {\text{id}}_A)=m\circ ({\text{id}}_A\otimes \iota )={\text{id}}_A\text{.}$

The relationship between the identity $1_A\in A$ and the unit $\iota :k\to A$ is $\iota \left(1\right)=1_A\text{.}$ If we are being precise we should denote an algebra over $k$ by a triple $(A,m,\iota )$ or $(A,m,1_A)$ but we shall usually be lazy and simply write $A\text{.}$

Definition of a module

Let $A$ be an algebra over $k\text{.}$ An $A\text{-module}$ is a vector space $M$ over $k$ with an $A\text{-action}$

$$\begin{array}{ccc}A\otimes M& ⟶& M\\ a\otimes m& ⟼& a·m=am\end{array}$$

such that

1. $\left(ab\right)m=a\left(bm\right),$ for all $a,b\in A$ and $m\in M,$ and
2. $1_{A}m=m,$ for all $m\in M\text{.}$

Let $M$ and $N$ be $A\text{-modules.}$ An $A\text{-module}$ morphism from $M$ to $N$ is a map $\phi :M\to N$ such that

$$\phi \left(am\right)=a\phi \left(m\right),\text{for all}a\in A\text{and}m\in M\text{.}$$

The set of $A\text{-module}$ morphisms from $M$ to $N$ is denoted ${\text{Hom}}_A(M,N)\text{.}$ An $A\text{-module}$ is finite dimensional if it is finite dimensional as a vector space over $k\text{.}$

Motivation for SRMCwMFFs

Our interest will be in special algebras for which the category of finite dimensional $A\text{-module}$ has a lot of nice structure. We want to be able to take the tensor product of two $A\text{-modules}$ and get a new $A\text{-module,}$ we want to be able to take the dual of an $A\text{-module}$ and get a new $A\text{-module}$ and we want to have a 1-dimensional "trivial" $A\text{-module.}$

Definition of SRMCwMFFs

Let $A$ be an algebra over $k\text{.}$ The category of finite dimensional $A\text{-modules}$ is a strict rigid monoidal category such that the forgetful functor is monoidal (a SRMCwMFF for short) if

1. For every pair $M,N$ of finite dimensional $A\text{-modules}$ there is a given $A\text{-module}$ structure on $M\otimes N,$
2. For every finite dimensional $A\text{-module}$ $M$ there is a given $A\text{-module}$ structure on $M^{*}={\text{Hom}}_k(M,k),$
3. There is a distinguished one-dimensional $A\text{-module}$ $1$ with a distinguished basis element $1\in 1,$

and the following conditions are satisfied:

1. For all finite dimensional $A\text{-modules}$ $M,$ $N,$ and $P,$ $$(M\otimes N)\otimes P=M\otimes (N\otimes P)$$ as $A\text{-modules*.}$
2. The maps $$\begin{array}{ccc}\begin{array}{ccc}1\otimes M& \stackrel{\sim }{⟶}& M\\ 1\otimes m& ⟼& m\end{array}& \text{and}& \begin{array}{ccc}M\otimes 1& \stackrel{\sim }{⟶}& M\\ m\otimes 1& ⟼& m\end{array}\end{array}$$ are $A\text{-module}$ isomorphisms.
3. For each finite dimensional $A\text{-module}$ $M,$ the maps $$\begin{array}{ccc}\begin{array}{ccc}{M}^{*}\otimes M& \stackrel{\sim }{⟶}& 1\\ \phi \otimes m& ⟼& \phi \left(m\right)·1\end{array}& \text{and}& \begin{array}{ccc}1& \stackrel{\sim }{⟶}& M\otimes M^{*}\\ 1& ⟼& {\sum }_i{m}_i\otimes {\phi }_i\end{array}\end{array}$$ are $A\text{-module}$ morphisms.

In condition (3) the set $\left\{m_i\right\}$ is a basis of $M$ and the set $\left\{{\phi }_i\right\}$ is the dual basis in $M^{*},$ i.e. ${\phi }_i\in M^{*}$ is such that ${\phi }_i\left(m_j\right)={\delta }_{ij}$ for all $i,j\text{.}$

The distinguished one-dimensional $A\text{-module}$ $1$ is called the trivial $A$ module.

* Strictly speaking we can only identify $(M\otimes N)\otimes P$ and $M\otimes (N\otimes P)$ up to coherent natural isomorphisms. If we are being precise this is crucial, but conceptually these two spaces are "equal".

Hopf algebras

Definition of Hopf algebras

A Hopf algebra is a vector space $A$ over $k$ with

$$\begin{array}{cc}\text{a multiplication,}& m:A\otimes A⟶A,\\ \text{a comultiplication,}& \Delta :A⟶A\otimes A,\\ \text{a unit,}& \iota :k⟶A,\\ \text{a counit,}& \epsilon :A⟶k,\text{and}\\ \text{an antipode,}& S:A⟶A,\end{array}$$

such that

1. $m$ is associative, $$m\circ ({\text{id}}_A\otimes m)=m\circ (m\otimes {\text{id}}_A),$$
2. $\Delta$ is coassociative, $$({\text{id}}_A\otimes \Delta )\circ \Delta =(\Delta \otimes {\text{id}}_A)\circ \Delta ,$$
3. (unit condition), $$m\circ ({\text{id}}_A\otimes \iota )=m\circ (\iota \otimes {\text{id}}_A)={\text{id}}_A,$$
4. (counit condition), $$({\text{id}}_A\otimes \epsilon )\circ \Delta =(\epsilon \otimes {\text{id}}_A)\circ \Delta ={\text{id}}_A,$$
5. $\Delta$ is an algebra homomorphism, $$\Delta \circ m=(m\otimes m)\circ ({\text{id}}_A\otimes \tau \otimes {\text{id}}_A)\circ (\Delta \otimes \Delta ),$$
6. $\epsilon$ is an algebra homomorphism, $$\epsilon \circ m=\epsilon \otimes \epsilon ,$$
7. (antipode condition), $$m\circ ({\text{id}}_A\otimes S)\circ \Delta =m\circ (S\otimes {\text{id}}_A)\circ \Delta =\iota \circ \epsilon \text{.}$$

In condition (5) the algebra structure on $A\otimes A$ is given by

$$(a\otimes b)(c\otimes d)=ac\otimes bd,\text{for all}a,b,c,d\in A,$$

and the map $\tau$ is given by

$$\begin{array}{cccc}\tau :& A\otimes A& ⟶& A\otimes A\\ & a\otimes b& ⟼& b\otimes a\text{.}\end{array}$$

In condition (6) we have identified the vector space $k\otimes k$ with $k\text{.}$ Once can show that the antipode $S:A\to A$ is always an anti-homomorphism,

Sweedler notation for the comultiplication

Let $A$ be a Hopf algebra over $k\text{.}$ If $a\in A$ we write

$$\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}$$

to express $\Delta \left(a\right)$ as an element of $A\otimes A\text{.}$ This unusual notation is called Sweedler notation and is a standard notation for working with Hopf algebras. Don't let it bother you, we are simply trying to write $\Delta \left(a\right)$ so that it looks like an element of $A\otimes A,$ without having to go through the rigmarole of actually choosing a basis in $A\text{.}$

Hopf algebras give us SRMCwMFFs!

Let $(A,m,\Delta ,\iota ,\epsilon ,S)$ be a Hopf algebra over $k\text{.}$

1. If $M_1$ and $M_2$ are $A\text{-modules}$ define an $A\text{-module}$ structure on $M_1\otimes M_2$ by $$a(m_1\otimes m_2)=\Delta \left(a\right)(m_1\otimes m_2)=\sum _a{a}_{\left(1\right)}{m}_1\otimes a_{\left(2\right)}{m}_2,$$ for each $a\in A,$ $m_1\in M_1,$ and $m_2\in M_2\text{.}$
2. Define $1$ to be the vector space $1=k·1$ and define an action of $A$ on $1$ by $$a·1=\epsilon \left(a\right)·1,\text{for each}a\in A\text{.}$$
3. If $M$ is a finite dimensional $A\text{-module}$ define an $A\text{-module}$ structure on $M^{*}={\text{Hom}}_k(M,k)$ by $$\left(a\phi \right)\left(m\right)=\phi \left(S\left(a\right)m\right),\text{for each}a\in A,\phi \in M^{*},\text{and}m\in M\text{.}$$

The point is that if $A$ is a Hopf algebra then, with the definitions in (a)-(c) above, the category of finite dimensional $A\text{-modules}$ is very nice; it is a strict monoidal category such that the forgetful functor is monoidal.

Group algebras are Hopf algebras

Let $G$ be a group. The group algebra of $G$ over $k$ is the vector space $kG$ of finite $k\text{-linear}$ combinations of elements of $G,$

$$kG=\{\sum _g{c}_{g}g\mid c_g\in k\text{and all but a finite number of}{c}_g=0\},$$

with multiplication given by the $k\text{-linear}$ extension of the multiplication in $G\text{.}$ a $G\text{-module}$ is a $kG\text{-module.}$

1. If $M_1$ and $M_2$ are $G\text{-modules}$ define a $G\text{-module}$ structure on $M_1\otimes M_2$ by $$g(m_1\otimes m_2)=g{m}_1\otimes g{m}_2,\text{for all}g\in G,m_1\in M_1,\text{and}{m}_2\in M_2\text{.}$$
2. The trivial $G\text{-module}$ is the 1-dimensional vector space $1$ with $G\text{-action}$ given by $$g·v=v,\text{for all}g\in G,v\in 1\text{.}$$
3. If $M$ is a finite dimensional $G\text{-module}$ define a $G\text{-module}$ structure on $M^{*}={\text{Hom}}_k(M,k)$ by $$\left(g\phi \right)\left(m\right)=\phi \left(g^{-1}m\right),\text{for all}g\in G,m\in M,\text{and}\phi \in M^{*}\text{.}$$

With these definitions the category of finite dimensional $G\text{-modules}$ is a strict monoidal category such that the forgetful functor is monoidal.

The group algebra $kG$ is a Hopf algebra if we define

1. a comultiplication, $\Delta :kG\to kG\otimes kG,$ by $$\Delta \left(g\right)=g\otimes g\text{for all}g\in G,$$
2. a counit, $\epsilon :kG\to k,$ by $$\epsilon \left(g\right)=1,\text{for all}g\in G,$$
3. and an antipode, $S:kG\to kG,$ by $$S\left(g\right)=g^{-1},\text{for all}g\in G\text{.}$$

Enveloping algebras of Lie algebras are Hopf algebras

Let $𝔤$ be a Lie algebra over $k$ and let $𝔘𝔤$ be its enveloping algebra. (See II (1.1) and II (4.2) for definitions of Lie algebras and enveloping algebras.)

1. If $M_1$ and $M_2$ are $𝔤\text{-modules}$ we define a $g\text{-module}$ structure on $M_1\otimes $ M_2$$ by $$x(m_1\otimes m_2)=x{m}_1\otimes m_2+m_1\otimes x{m}_2,\text{for all}x\in 𝔤,m_1\in M_1,\text{and}{m}_2\in M_2\text{.}$$
2. The trivial $𝔤\text{-module}$ is the 1-dimensional vector space $1$ with $𝔤\text{-action}$ given by $$xv=0,\text{for all}x\in 𝔤,v\in 1\text{.}$$
3. If $M$ is a finite dimensional $𝔤\text{-module}$ we define a $𝔤\text{-module}$ structure on $M^{*}={\text{Hom}}_k(M,k)$ by $$\left(x\phi \right)\left(m\right)=\phi (-xm),\text{for all}x\in 𝔤,\phi \in M^{*},\text{and}m\in M\text{.}$$

With these definitions the category of finite dimensional $𝔤\text{-modules}$ is a strict rigid monoidal category such that the forgetful functor is monoidal.

The enveloping algebra $𝔘𝔤$ of $𝔤$ is a Hopf algebra if we define

1. a comultiplication, $\Delta :𝔘𝔤\to 𝔘𝔤\otimes 𝔘𝔤,$ by $$\Delta \left(x\right)=x\otimes 1+1\otimes x,\text{for all}x\in 𝔤,$$
2. a counit, $\epsilon :𝔘𝔤\to k,$ by $$\epsilon \left(x\right)=0,\text{for all}x\in 𝔤,$$
3. and an antipode, $S:𝔘𝔤\to 𝔘𝔤,$ by $$S\left(x\right)=-x,\text{for all}x\in 𝔤\text{.}$$

Definition of the adjoint action of a Hopf algebra on itself

Let $(A,m,\Delta ,\iota ,\epsilon ,S)$ be a Hopf algebra. The vector space $A$ is an $A\text{-module}$ where the action of $A$ on $A$ is given by

$$\begin{array}{ccc}A\otimes A& ⟶& A\\ a\otimes b& ⟼& {\sum }_a{a}_{\left(1\right)}bS\left(a_{\left(2\right)}\right)\end{array},\text{where}\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}\text{.}$$

The linear transformation of $A$ determined by the action of an element $a\in A$ is denoted ${\text{ad}}_a\text{.}$ Thus,

$${\text{ad}}_a\left(b\right)=\sum _a{a}_{\left(1\right)}bs\left(a_{\left(2\right)}\right),\text{for all}b\in A\text{.}$$

Motivation for the definition of the adjoint action

Let $M$ be an $A\text{-module}$ and let $\rho :A\to \text{End}\left(M\right)$ be the corresponding representation of $A,$ i.e. the map

$$\begin{array}{cccc}\rho :& A& ⟶& \text{End}\left(M\right)\\ & a& ⟼& \rho \left(a\right)\end{array}$$

where $\rho \left(a\right)$ is the linear tranformation of $M$ determined by the action of $a\text{.}$ Note that $\text{End}\left(M\right)\cong M\otimes M^{*}$ as a vector space. On the other hand $M\otimes M^{*}$ is an $A\text{-module.}$ If we view $A$ as an $A\text{-module}$ under the adjoint action then the composite map

$$\rho :A⟶\text{End}\left(M\right)\cong M\otimes M^{*}$$

is a homomorphism of $A\text{-modules.}$

Definition of an ad-invariant bilinear form on a Hopf algebra

Let $A$ be a Hopf algebra with antipode $S$ and let $M$ be an $A\text{-module.}$ A bilinear form

$$\begin{array}{cccc}(,):& M\otimes M& ⟶& k\\ & m\otimes n& ⟼& (m,n)\end{array}\text{is}\text{invariant}\text{if}(a{m}_1,m_2)=(m_1,S\left(a\right)m_2),$$

for all $a\in A,$ $m_1,m_2\in M\text{.}$ This is equivalent to the condition that the map $(,)$ is a homomorphism of $A\text{-modules}$ when we identify $k$ with the trivial $A\text{-module}$ $1\text{.}$

A bilinear form

$$(,):A\otimes A⟶k\text{is}\text{ad-invariant}\text{if}({\text{ad}}_a\left(b_1\right),b_2)=(b_1,{\text{ad}}_{S\left(a\right)}\left(b_2\right)),$$

for all $a,b_1,b_2\in A\text{.}$ In other words, the bilinear form is invariant if we view $A$ as an $A\text{-module}$ via the adjoint action.

Braided SRMCwMFFs

Motivation for braided SRMCwMFFs

Our interest here will be in even more special algebras for which the category of finite dimensional $A\text{-modules}$ is "braided". Specifically, we want the two tensor product modules $M\otimes N$ and $N\otimes M$ to be isomorphic.

Definition of braided SRMCwMFFs

Let $A$ be an algebra over $k\text{.}$ The category of finite dimensional $A\text{-modules}$ is a braided strict rigid monoidal category such that the forgetful functor is monoidal (a braided SRMCwMFF for short) if it is a strict rigid monoidal category such that the forgetful functor is monoidal and

1. There is a family of braiding isomorphisms $${\stackrel{\vee }{R}}_{M,N}:M\otimes N⟶N\otimes M,$$ which are natural isomorphisms (in the sense of the theory of categories).
2. For all finite dimensional $A\text{-modules}$ $M,N,P$ $$\begin{array}{c}{\stackrel{\vee }{R}}_{M\otimes N,P}=({\stackrel{\vee }{R}}_{M\otimes P}\otimes {\text{id}}_N)\circ ({\text{id}}_M\otimes {\stackrel{\vee }{R}}_{N,P}),\\ {\stackrel{\vee }{R}}_{M,N\otimes P}=({\text{id}}_N\otimes {\stackrel{\vee }{R}}_{M\otimes P})\circ ({\stackrel{\vee }{R}}_{M\otimes N}\otimes {\text{id}}_P),\text{and}\\ {\stackrel{\vee }{R}}_{1,M}={\text{id}}_M={\stackrel{\vee }{R}}_{M,1}\end{array}$$ where $1$ denotes the trivial module and we identify $M,$ $1\otimes M,$ and $M\otimes 1\text{.}$

Pictorial representation of braiding isomorphisms

Sometimes it is convenient to denote the isomorphism ${\stackrel{\vee }{R}}_{M\otimes N}:M\otimes N⟶N\otimes M$ by the picture

$$M ⊗ N N ⊗ M$$

With this notation the relations defining a braided SRMCwMFF can be written in the form

$$\begin{array}{ccc} (M⊗N)⊗P P⊗(M⊗N) & =& M ⊗ N ⊗ P P ⊗ N ⊗ M \\ M⊗(N⊗P) (N⊗P)⊗M & =& M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}$$ $$\begin{array}{ccccc} 1⊗M M⊗1 & =& M M & =& 1⊗M M⊗1 \end{array}$$

What "natural isomorphism" means

Let $M,M^{\prime },N,N^{\prime }$ be $A\text{-modules}$ and let $\tau :M\to M^{\prime }$ and $\sigma :N\to N^{\prime }$ be $A\text{-module}$ isomorphisms. Then the naturality condition on the isomorphisms ${\stackrel{\vee }{R}}_{M,N}$ means that the following diagrams commute.

$$\begin{array}{cc}\begin{array}{ccc}M\otimes N& \stackrel{\tau \otimes {\text{id}}_N}{⟶}& M^{\prime }\otimes N\\ {\stackrel{\vee }{R}}_{M,N}\downarrow & & \downarrow {\stackrel{\vee }{R}}_{M^{\prime },N}\\ N\otimes M& \stackrel{{\text{id}}_N\otimes \tau }{⟶}& N\otimes M^{\prime }\end{array}& \begin{array}{ccc}M\otimes N& \stackrel{{\text{id}}_M\otimes \sigma }{⟶}& M\otimes N^{\prime }\\ {\stackrel{\vee }{R}}_{M,N}\downarrow & & \downarrow {\stackrel{\vee }{R}}_{M,N^{\prime }}\\ N\otimes M& \stackrel{\sigma \otimes {\text{id}}_M}{⟶}& N^{\prime }\otimes M\end{array}\end{array}$$

Pictorially we have

$$\begin{array}{ccc}\begin{array}{ccc} M⊗N M′⊗N N⊗M′ τ & =& M⊗N N⊗M N⊗M′ τ \end{array}& \text{and}& \begin{array}{ccc} M⊗N M⊗N′ N′⊗M σ & =& M⊗N N⊗M N′⊗M σ \end{array}\end{array}$$

The braid relation

The relations in (3.3) imply the following relation which is usually called the braid relation.

$$\begin{array}{ccccccc} M ⊗ N ⊗ P P ⊗ N ⊗ M & =& M⊗N⊗P M⊗(P⊗N) (P⊗N)⊗M & =& M⊗(N⊗P) (N⊗P)⊗M P⊗N⊗M & =& M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}$$

where the middle equality is a consequence of the naturality property and the fact that the map ${\stackrel{\vee }{R}}_{N,P}$ is an isomorphism.

Quasitriangular Hopf algebras

Motivation for quasitriangular Hopf algebras

In addition to the definition of a braided SRMCwMFF the following observations help to motivate the definition of a quasitriangular Hopf algebra.

Let $(A,m,\Delta ,\iota ,\epsilon ,S)$ be a Hopf algebra and let $\tau$ be the $k\text{-linear}$ map

$$\begin{array}{cccc}\tau :& A\otimes A& ⟶& A\otimes A\\ & a\otimes b& ⟼& b\otimes a\text{.}\end{array}$$

Let ${\Delta }^{\text{op}}=\tau \circ \Delta$ so that, if $a\in A$ and

$$\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)},\text{then}{\Delta }^{\text{op}}\left(a\right)=\sum _a{a}_{\left(2\right)}\otimes a_{\left(1\right)}\text{.}$$

Then $(A,m,{\Delta }^{\text{op}},\iota ,\epsilon ,S^{-1})$ is a Hopf algebra.

The map $\tau :A\otimes A\to A\otimes A$ is an algebra automorphism of $A\otimes A$ (the algebra structure on $A\otimes A$ is as given in (2.1)) and the following diagram commutes

$$\begin{array}{ccc}A& \stackrel{\Delta }{⟶}& A\otimes A\\ {\text{id}}_A\downarrow & & \downarrow \tau \\ A& \stackrel{{\Delta }^{\text{op}}}{⟶}& A\otimes A\end{array}$$

Sometimes we are lucky and we can replace $\tau$ by an inner automorphism.

Definition of quasitriangular Hopf algebras

A quasitriangular Hopf algebra is a pair $(A,ℛ)$ where $A$ is a Hopf algebra and $ℛ$ is an invertible element of $A\otimes A$ such that

$$\begin{array}{c}{\Delta }^{\text{op}}\left(a\right)=ℛ\Delta \left(a\right){ℛ}^{-1},\text{for all}a\in A,\text{and}\\ (\Delta \otimes {\text{id}}_A)\left(ℛ\right)={ℛ}_{13}{ℛ}_{23},\text{and}({\text{id}}_A\otimes \Delta )\left(ℛ\right)={ℛ}_{13}{ℛ}_{12},\end{array}$$

where, if $ℛ=\sum a_i\otimes b_i,$ then

$${ℛ}_{12}=\sum a_i\otimes b_i\otimes 1,{ℛ}_{13}=\sum a_i\otimes 1b_i,\text{and}{ℛ}_{23}=\sum 1\otimes a_i\otimes b_i\text{.}$$

Quasitriangular Hopf algebras give braided SRMCwMFFs

Let $(A,ℛ)$ be a quasitriangular Hopf algebra. For each pair of finite dimensional $A\text{-modules}$ $M,N$ define

$$\begin{array}{cccc}{\stackrel{\vee }{R}}_{M,N}:& M\otimes N& ⟶& N\otimes M\\ & m\otimes n& ⟼& \sum b_{i}n\otimes a_{i}m,\end{array}$$

where $ℛ=\sum a_i\otimes b_i\in A\otimes A\text{.}$ Then the category of finite dimensional $A\text{-modules}$ is a braided strict monoidal category such that the forgetful functor is monoidal.

The quantum double

Motivation for the quantum double

In general it can be very difficult to find quasitriangular Hopf algebras, especially ones where the element $ℛ$ is different from $1\otimes 1\text{.}$ The construction in (5.3) 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.

Construction of the Hopf algebra $A^{*\text{coop}}$

Let $(A,m,\Delta ,\iota ,\epsilon ,S)$ be a finite dimensional Hopf algebra over $k\text{.}$ Let $A^{*}={\text{Hom}}_k(A,k)$ be the dual of $A\text{.}$ There is a natural bilinear pairing $⟨,⟩:A^{*}\otimes A⟶k$ between $A$ and $A^{*}$ given by

$$⟨\alpha ,a⟩=\alpha \left(a\right),\text{for all}\alpha \in A^{*}\text{and}a\in A\text{.}$$

Extend this notation so that if ${\alpha }_1,{\alpha }_2\in A^{*}$ and $a_1,a_2\in A$ then

$$⟨{\alpha }_1\otimes {\alpha }_2,{\alpha }_1\otimes {\alpha }_2⟩=⟨{\alpha }_1,a_1⟩⟨{\alpha }_2,a_2⟩\text{.}$$

We make $A^{*}$ into a Hopf algebra, which is denoted $A^{*\text{coop}},$ by defining a multiplication and a comultiplication $\Delta$ on $A^{*}$ via equations

$$⟨{\alpha }_1{\alpha }_2,a⟩=⟨{\alpha }_1\otimes {\alpha }_2,\Delta \left(a\right)⟩\text{and}⟨{\Delta }^{\text{op}}\left(\alpha \right),a_1\otimes a_2⟩-⟨\alpha ,a_1{a}_2⟩,$$

for all $\alpha ,{\alpha }_1,{\alpha }_2\in A^{*}$ and $a,a_1,a_2\in A\text{.}$ The definition of ${\Delta }^{\text{op}}$ is in (4.1).

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

Construction of the quantum double

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

1. We paste $A$ and $A^{*\text{coop}}$ together by letting $$D\left(A\right)=A\otimes A^{*\text{coop}}\text{.}$$ Write elements of $D\left(A\right)$ as $a\alpha$ instead of as $a\otimes \alpha \text{.}$
2. We want the multiplication in $D\left(A\right)$ to reflect the multiplication in $A$ and the multiplication in $A^{*\text{coop}}\text{.}$ Similarly for the comultiplication.
3. We want the $ℛ\text{-matrix}$ 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 $A^{*}\text{.}$
   The condition in (2) determines the comultiplication in $D\left(A\right),$ $$\Delta \left(\alpha a\right)=\Delta \left(\alpha \right)\Delta \left(a\right)=\sum _{a,\alpha }{a}_{\left(1\right)}{\alpha }_{\left(1\right)}\otimes a_{\left(2\right)}{\alpha }_{\left(2\right)},$$

where $\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}$ and $\Delta \left(\alpha \right)=\sum _{\alpha }{\alpha }_{\left(1\right)}\otimes {\alpha }_{\left(2\right)}\text{.}$ The condition in (2) doesn't quite determine the multiplication in $D\left(A\right)\text{.}$ We need to be able to expand products like $\left(a_1{\alpha }_1\right)\left(a_2{\alpha }_2\right)\text{.}$ If we knew

$${\alpha }_1{a}_2=\sum _j{b}_j{\beta }_j,\text{for some elements}{\beta }_j\in A^{*\text{coop}}\text{and}{b}_j\in A,$$

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 $D\left(A\right)\text{.}$ Miraculously, the condition in (3) and the equation

$$ℛ\Delta \left(a\right){ℛ}^{-1}={\Delta }^{\text{op}}\left(a\right),\text{for all}a\in A,$$

force that if $\alpha \in A^{*\text{coop}}$ and $a\in A$ then, in $D\left(A\right),$

$$\begin{array}{ccc}\alpha a& =& \sum _{\alpha ,a}⟨{\alpha }_{\left(1\right)},S^{-1}\left(a_{\left(1\right)}\right)⟩⟨{\alpha }_{\left(3\right)},a_{\left(3\right)}⟩a_{\left(2\right)}{\alpha }_{\left(2\right)},\text{and}\\ a\alpha & =& \sum _{\alpha ,a}⟨{\alpha }_{\left(1\right)},a_{\left(1\right)}⟩⟨{\alpha }_{\left(3\right)},S^{-1}\left(a_{\left(3\right)}\right)⟩{\alpha }_{\left(2\right)}{a}_{\left(2\right)},\end{array}$$

where, if $\Delta$ is the comultiplcation in $D\left(A\right),$

$$(\Delta \otimes \text{id})\circ \Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}\otimes a_{\left(3\right)},\text{and}(\Delta \otimes \text{id})\circ \Delta \left(\alpha \right)=\sum _{\alpha }{\alpha }_{\left(1\right)}\otimes {\alpha }_{\left(2\right)}\otimes {\alpha }_{\left(3\right)}\text{.}$$

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

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

1. The $k\text{-linear}$ map $$\begin{array}{ccc}A\otimes A^{*\text{coop}}& ⟶& D\left(A\right)\\ a\otimes \alpha & ⟼& a\alpha \end{array}$$ is bijective.
2. $D\left(A\right)$ contains $A$ and $A^{*\text{coop}}$ 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 dual basis in $A^{*\text{coop}}\text{.}$

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

If $A$ is an infinite dimensional Hopf algebra

It is sometimes possible to do an analogous construction when $A$ is infinite dimensional if one is careful about what the dual of $A$ is and how to express the (now infinite) sum $ℛ=\sum _i{b}_i\otimes b^i\text{.}$ To get an idea of how this is done see VII (7.1) and [Lus1993] Chapt. 4.

An ad-invariant pairing on the quantum double

Let $(A,m,\Delta ,\iota ,\epsilon ,S)$ be a Hopf algebra. The bilinear form on the quantum double $D\left(A\right)$ of $A$ which is defined by

$$⟨a\alpha ,b\beta ⟩=⟨\beta ,S\left(a\right)⟩⟨\alpha ,S^{-1}\left(b\right)⟩,\text{for all}a,b\in A\text{and all}\alpha ,\beta \in A^{*\text{coop}},$$

satisfies

$$⟨{\text{ad}}_u\left(x\right),y⟩=⟨x,{\text{ad}}_{S\left(u\right)}\left(y\right)⟩,\text{for all}u,x,y\in D\left(A\right)\text{.}$$

The proposition says that the bilinear form is ad-invariant, as defined in (2.8). The bilinear form is not necessarily symmetric,

$$⟨y,x⟩=⟨x,S^2\left(y\right)⟩,\text{for all}x,y\in D\left(A\right)\text{.}$$

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

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