## I. Hopf algebras and quasitriangular Hopf algebras

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 ${\text{id}}_{V}:\phantom{\rule{0.2em}{0ex}}V\to V$ 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

$m: A⊗A ⟶ A a⊗b ⟼ a·b=ab$

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 $m:\phantom{\rule{0.2em}{0ex}}A\otimes A\to A$ and a unit $\iota :\phantom{\rule{0.2em}{0ex}}k\to A$ such that

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

The relationship between the identity ${1}_{A}\in A$ and the unit $\iota :\phantom{\rule{0.2em}{0ex}}k\to A$ is $\iota \left(1\right)={1}_{A}\text{.}$ If we are being precise we should denote an algebra over $k$ by a triple $\left(A,m,\iota \right)$ or $\left(A,m,{1}_{A}\right)$ 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 $A\text{-action}$

$A⊗M ⟶ M a⊗m ⟼ a·m=am$

such that

1. $\left(ab\right)m=a\left(bm\right),\phantom{\rule{1em}{0ex}}$ for all $a,b\in A$ and $m\in M,$ and
2. ${1}_{A}m=m,\phantom{\rule{1em}{0ex}}$ 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 $\phi :\phantom{\rule{0.2em}{0ex}}M\to N$ such that

$φ(am)=aφ(m), for alla∈Aand m∈M.$

The set of $A\text{-module}$ morphisms from $M$ to $N$ is denoted ${\text{Hom}}_{A}\phantom{\rule{0.2em}{0ex}}\left(M,N\right)\text{.}$ 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 ${M}^{*}={\text{Hom}}_{k}\phantom{\rule{0.2em}{0ex}}\left(M,k\right),$
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 $P,$ $(M⊗N)⊗P= M⊗(N⊗P)$ as $A\text{-modules*.}$
2. The maps $1⊗M ⟶∼ M 1⊗m ⟼ m and M⊗1 ⟶∼ M m⊗1 ⟼ m$ are $A\text{-module}$ isomorphisms.
3. For each finite dimensional $A\text{-module}$ $M,$ the maps $M*⊗M ⟶∼ 1 φ⊗m ⟼ φ(m)·1 and 1 ⟶∼ M⊗M* 1 ⟼ ∑imi⊗φi$ 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 $\left(M\otimes N\right)\otimes P$ and $M\otimes \left(N\otimes P\right)$ 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

$a multiplication, m:A⊗A⟶A, a comultiplication, Δ:A⟶A⊗A, a unit, ι:k⟶A, a counit, ε:A⟶k,and an antipode, S:A⟶A,$

such that

1. $m$ is associative, $m∘(idA⊗m)= m∘(m⊗idA),$
2. $\Delta$ is coassociative, $(idA⊗Δ) ∘Δ= (Δ⊗idA) ∘Δ,$
3. (unit condition), $m∘(idA⊗ι)= m∘(ι⊗idA)= idA,$
4. (counit condition), $(idA⊗ε)∘ Δ= (ε⊗idA)∘ Δ=idA,$
5. $\Delta$ is an algebra homomorphism, $Δ∘m= (m⊗m)∘ (idA⊗τ⊗idA) ∘(Δ⊗Δ) ,$
6. $\epsilon$ is an algebra homomorphism, $ε∘m=ε⊗ε,$
7. (antipode condition), $m∘(idA⊗S)∘ Δ=m∘ (S⊗idA)∘ Δ=ι∘ε.$

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

$(a⊗b) (c⊗d)= ac⊗bd, for alla,b,c,d∈A,$

and the map $\tau$ is given by

$τ: A⊗A ⟶ A⊗A a⊗b ⟼ b⊗a.$

In condition (6) we have identified the vector space $k\otimes k$ with $k\text{.}$ Once can show that the antipode $S:\phantom{\rule{0.2em}{0ex}}A\to A$ is always an anti-homomorphism,

$S(ab)=S(b) S(a),for all a,b∈A.$

### Sweedler notation for the comultiplication

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

$Δ(a)=∑a a(1)⊗a(2)$

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 $\left(A,m,\Delta ,\iota ,\epsilon ,S\right)$ 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 $a(m1⊗m2)= Δ(a) (m1⊗m2)= ∑aa(1)m1 ⊗a(2)m2,$ 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 $a·1=ε(a)·1, for eacha∈A.$
3. If $M$ is a finite dimensional $A\text{-module}$ define an $A\text{-module}$ structure on ${M}^{*}={\text{Hom}}_{k}\phantom{\rule{0.2em}{0ex}}\left(M,k\right)$ by $(aφ)(m)=φ (S(a)m), for eacha∈A, φ∈M*,and m∈M.$

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 $G,$

$kG= { ∑gcgg∣ cg∈kand all but a finite number of cg=0 } ,$

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 $g(m1⊗m2)= gm1⊗gm2, for allg∈G, m1∈M1,and m2∈M2.$
2. The trivial $G\text{-module}$ is the 1-dimensional vector space $1$ with $G\text{-action}$ given by $g·v=v,for all g∈G,v∈1.$
3. If $M$ is a finite dimensional $G\text{-module}$ define a $G\text{-module}$ structure on ${M}^{*}={\text{Hom}}_{k}\phantom{\rule{0.2em}{0ex}}\left(M,k\right)$ by $(gφ)(m)= φ(g-1m), for allg∈G,m∈M, andφ∈M*.$

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, $\phantom{\rule{1em}{0ex}}\Delta :\phantom{\rule{0.2em}{0ex}}kG\to kG\otimes kG,$ by $Δ(g)=g⊗g for allg∈G,$
2. a counit, $\phantom{\rule{1em}{0ex}}\epsilon :\phantom{\rule{0.2em}{0ex}}kG\to k,$ by $ε(g)=1,for all g∈G,$
3. and an antipode, $\phantom{\rule{1em}{0ex}}S:\phantom{\rule{0.2em}{0ex}}kG\to kG,$ by $S(g)=g-1, for allg∈G.$

### 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 $x(m1⊗m2)=x m1⊗m2+m1⊗ xm2,for allx∈ 𝔤,m1∈M1, andm2∈M2.$
2. The trivial $𝔤\text{-module}$ is the 1-dimensional vector space $1$ with $𝔤\text{-action}$ given by $xv=0,for allx∈𝔤 ,v∈1.$
3. If $M$ is a finite dimensional $𝔤\text{-module}$ we define a $𝔤\text{-module}$ structure on ${M}^{*}={\text{Hom}}_{k}\phantom{\rule{0.2em}{0ex}}\left(M,k\right)$ by $(xφ)(m)=φ (-xm),for all x∈𝔤,φ∈M*, andm∈M.$

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, $\phantom{\rule{1em}{0ex}}\Delta :\phantom{\rule{0.2em}{0ex}}𝔘𝔤\to 𝔘𝔤\otimes 𝔘𝔤,$ by $Δ(x)=x⊗1+ 1⊗x,for allx∈𝔤,$
2. a counit, $\phantom{\rule{1em}{0ex}}\epsilon :\phantom{\rule{0.2em}{0ex}}𝔘𝔤\to k,$ by $ε(x)=0,for all x∈𝔤,$
3. and an antipode, $\phantom{\rule{1em}{0ex}}S:\phantom{\rule{0.2em}{0ex}}𝔘𝔤\to 𝔘𝔤,$ by $S(x)=-x,for all x∈𝔤.$

### Definition of the adjoint action of a Hopf algebra on itself

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

$A⊗A ⟶ A a⊗b ⟼ ∑aa(1)bS(a(2)) ,whereΔ(a) =∑aa(1)⊗ a(2).$

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

$ada(b)=∑a a(1)bs (a(2)), for allb∈A.$

### Motivation for the definition of the adjoint action

Let $M$ be an $A\text{-module}$ and let $\rho :\phantom{\rule{0.2em}{0ex}}A\to \text{End}\phantom{\rule{0.2em}{0ex}}\left(M\right)$ be the corresponding representation of $A,$ i.e. the map

$ρ: A ⟶ End(M) a ⟼ ρ(a)$

where $\rho \left(a\right)$ is the linear tranformation of $M$ determined by the action of $a\text{.}$ Note that $\text{End}\phantom{\rule{0.2em}{0ex}}\left(M\right)\cong M\otimes {M}^{*}$ 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

$ρ:A⟶End(M) ≅M⊗M*$

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

$(,): M⊗M ⟶ k m⊗n ⟼ (m,n) isinvariant if(am1,m2) =(m1,S(a)m2) ,$

for all $a\in A,$ ${m}_{1},{m}_{2}\in M\text{.}$ This is equivalent to the condition that the map $\left(,\right)$ is a homomorphism of $A\text{-modules}$ when we identify $k$ with the trivial $A\text{-module}$ $1\text{.}$

A bilinear form

$(,):A⊗A⟶k isad-invariantif (ada(b1),b2) =(b1,adS(a)(b2)) ,$

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 $R∨M,N: M⊗N⟶N⊗M,$ which are natural isomorphisms (in the sense of the theory of categories).
2. For all finite dimensional $A\text{-modules}$ $M,N,P$ $R∨M⊗N,P= ( R∨M⊗P ⊗idN ) ∘ ( idM⊗ R∨N,P ) , R∨M,N⊗P= ( idN⊗ R∨M⊗P ) ∘ ( R∨M⊗N ⊗idP ) ,and R∨1,M= idM= R∨M,1$ 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 ${\stackrel{\vee }{R}}_{M\otimes N}:\phantom{\rule{0.2em}{0ex}}M\otimes N⟶N\otimes M$ by the picture

$M ⊗ N N ⊗ M$

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

$(M⊗N)⊗P P⊗(M⊗N) = M ⊗ N ⊗ P P ⊗ N ⊗ M M⊗(N⊗P) (N⊗P)⊗M = M ⊗ N ⊗ P P ⊗ N ⊗ M$ $1⊗M M⊗1 = M M = 1⊗M M⊗1$

### What "natural isomorphism" means

Let $M,{M}^{\prime },N,{N}^{\prime }$ be $A\text{-modules}$ and let $\tau :\phantom{\rule{0.2em}{0ex}}M\to {M}^{\prime }$ and $\sigma :\phantom{\rule{0.2em}{0ex}}N\to {N}^{\prime }$ be $A\text{-module}$ isomorphisms. Then the naturality condition on the isomorphisms ${\stackrel{\vee }{R}}_{M,N}$ means that the following diagrams commute.

$M⊗N ⟶τ⊗idN M′⊗N R∨M,N ↓ ↓ R∨M′,N N⊗M ⟶idN⊗τ N⊗M′ M⊗N ⟶idM⊗σ M⊗N′ R∨M,N ↓ ↓ R∨M,N′ N⊗M ⟶σ⊗idM N′⊗M$

Pictorially we have

$M⊗N M′⊗N N⊗M′ τ = M⊗N N⊗M N⊗M′ τ and M⊗N M⊗N′ N′⊗M σ = M⊗N N⊗M N′⊗M σ$

### The braid relation

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

$M ⊗ N ⊗ P P ⊗ N ⊗ M = M⊗N⊗P M⊗(P⊗N) (P⊗N)⊗M = M⊗(N⊗P) (N⊗P)⊗M P⊗N⊗M = M ⊗ N ⊗ P P ⊗ N ⊗ M$

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

$τ: A⊗A ⟶ A⊗A a⊗b ⟼ b⊗a.$

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

$Δ(a)=∑a a(1)⊗a(2), thenΔop (a)=∑a a(2)⊗a(1).$

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

The map $\tau :\phantom{\rule{0.2em}{0ex}}A\otimes A\to A\otimes A$ 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

$A ⟶Δ A⊗A idA ↓ ↓ τ A ⟶Δop A⊗A$

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

$Δop(a)= ℛΔ(a) ℛ-1,for all a∈A,and (Δ⊗idA) (ℛ)=ℛ13 ℛ23,and (idA⊗Δ) (ℛ)=ℛ13 ℛ12,$

where, if $ℛ=\sum {a}_{i}\otimes {b}_{i},$ then

$ℛ12=∑ai⊗ bi⊗1,ℛ13 =∑ai⊗1bi,and ℛ23=∑1⊗ ai⊗bi.$

### Quasitriangular Hopf algebras give braided SRMCwMFFs

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

$R∨M,N: M⊗N ⟶ N⊗M m⊗n ⟼ ∑bin⊗aim,$

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 $\left(A,m,\Delta ,\iota ,\epsilon ,S\right)$ be a finite dimensional Hopf algebra over $k\text{.}$ Let ${A}^{*}={\text{Hom}}_{k}\phantom{\rule{0.2em}{0ex}}\left(A,k\right)$ be the dual of $A\text{.}$ There is a natural bilinear pairing $⟨,⟩:\phantom{\rule{0.2em}{0ex}}{A}^{*}\otimes A⟶k$ between $A$ and ${A}^{*}$ given by

$⟨α,a⟩=α (a),for all α∈A*anda∈ A.$

Extend this notation so that if ${\alpha }_{1},{\alpha }_{2}\in {A}^{*}$ and ${a}_{1},{a}_{2}\in A$ then

$⟨ α1⊗α2,α1 ⊗α2 ⟩ =⟨α1,a1⟩ ⟨α2,a2⟩.$

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

$⟨α1α2,a⟩= ⟨ α1⊗α2, Δ(a) ⟩ and ⟨ Δop(α), a1⊗a2 ⟩ -⟨α,a1a2⟩ ,$

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 $\epsilon :\phantom{\rule{0.2em}{0ex}}A\to k$ of $A\text{.}$
2. The counit of ${A}^{*\text{coop}}$ is the map $ε: A* → k α ↦ a(1),$ 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 $D(A)=A⊗ A*coop.$ 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 $ℛ=∑ibi⊗ bi,$ 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 $D\left(A\right),$ $Δ(αa)= Δ(α) Δ(a)= ∑a,α a(1) α(1)⊗ a(2) α(2),$

where $\Delta \left(a\right)=\sum _{a}{a}_{\left(1\right)}\otimes {a}_{\left(2\right)}$ and $\Delta \left(\alpha \right)=\sum _{\alpha }{\alpha }_{\left(1\right)}\otimes {\alpha }_{\left(2\right)}\text{.}$ 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

$α1a2=∑j bjβj, for some elementsβj∈ A*coopand bj∈A,$

then we would have

$(a1α1) (a2α2) =∑j (a1bj) (βjα2)$

which is a well defined element of $D\left(A\right)\text{.}$ Miraculously, the condition in (3) and the equation

$ℛΔ(a) ℛ-1= Δop(a) ,for alla∈A,$

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

$αa = ∑α,a ⟨ α(1),S-1 (a(1)) ⟩ ⟨ α(3), a(3) ⟩ a(2) α(2), and aα = ∑α,a ⟨ α(1), a(1) ⟩ ⟨ α(3),S-1 (a(3)) ⟩ α(2)a(2),$

where, if $\Delta$ is the comultiplcation 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 $D\left(A\right)\text{.}$ 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 ${A}^{*}={\text{Hom}}_{k}\phantom{\rule{0.2em}{0ex}}\left(A,k\right)$ except with opposite comultiplication. Then there exists a unique quasitriangular Hopf algebra $\left(D\left(A\right),ℛ\right)$ given by

1. The $k\text{-linear}$ map $A⊗A*coop ⟶ D(A) a⊗α ⟼ aα$ 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 $ℛ=∑ibi⊗bi,$ 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 $\left(A,m,\Delta ,\iota ,\epsilon ,S\right)$ be a Hopf algebra. The bilinear form on the quantum double $D\left(A\right)$ of $A$ which is defined by

$⟨aα,bβ⟩= ⟨β,S(a)⟩ ⟨α,S-1(b)⟩, for alla,b∈A and allα,β∈ A*coop,$

satisfies

$⟨adu(x),y⟩ =⟨x,adS(u)(y)⟩ ,for allu,x,y∈D(A).$

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

$⟨y,x⟩= ⟨x,S2(y)⟩, for allx,y∈D(A).$

## 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.