Hopf Algebras

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: 1 April 2010

Hopf algberas

Let $𝕂$ be a commutative ring. A vector space over $𝕂$ is a free $𝕂$-module. Unless otherwise specified all maps between vector spaces over $𝕂$ are assumed to be $𝕂$-linear and, if $V$ is a vector space over $𝕂$, then $\mathrm{id}:V\to V$ denotes the identity map from $V$ to $V.$

An algebra over $𝕂$ is a vector space over $𝕂$ with a multiplication and an identity element $1\in A$ such that

1. $m$ is associative, ie $a_1\left(a_2{a}_3\right)=\left(a_1{a}_2\right)a_3,$ for all $a_1,a_2,a_3\in A,$ and
2. $1a=a1=a,$ for all $a\in A.$

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

1. $m$ is associative, ie $m\left(m\otimes \mathrm{id}\right)=m\left(\mathrm{id}\otimes m\right),$ and
2. (unit condition) $m\left(i\otimes \mathrm{id}\right)=m\left(\mathrm{id}\otimes i\right)=\mathrm{id}.$

The relationship between the identity $1\in A$ and the unit $i:𝕂\to A$ is $i\left(1\right)=1.$

Let $A$ be an algbera over $𝕂.$ An $A$-module is a vector space $A$ over $𝕂$ with an $A$-action $$\begin{array}{rcl}A\otimes M& \to & M\\ a\otimes m& \mapsto & am\end{array}\text{such that}\left(a_1{a}_2\right)m=a_1\left(a_{2}m\right),\text{and}1m=m,$$ for all $a_1,a_2\in A$ and $m\in M.$

Let $M$ and $N$ be $A$-modules. An $A$-module morphism from $M$ to $N$ is a map $\phi :M\to N$ such that $$\phi \left(am\right)=a\phi \left(m\right),\text{for all}a\in A\text{and}m\in M.$$ The set of $A$-module morphisms from $M$ to $N$ is denoted ${\mathrm{Hom}}_A\left(M,N\right).$

A Hopf algebra is a vector space $A$ over $𝕂$ with such that

1. $m$ is associative, $m\left(\mathrm{id}\otimes m\right)=m\left(m\otimes \mathrm{id}\right),$
2. $\Delta$ is coassociative, $\left(\mathrm{id}\otimes \Delta \right)\Delta =\left(\Delta \otimes \mathrm{id}\right)\Delta ,$
3. (unit condition), $m\left(\mathrm{id}\otimes i\right)=m\left(i\otimes \mathrm{id}\right)=\mathrm{id},$
4. (counit condition), $\left(\mathrm{id}\otimes \epsilon \right)\Delta =\left(\epsilon \otimes \mathrm{id}\right)\Delta =\mathrm{id},$
5. $\Delta$ is an algebra homomorphism, $\Delta m=\left(m\otimes m\right)\left(\mathrm{id}\otimes \tau \otimes \mathrm{id}\right)\left(\Delta \otimes \Delta \right),$
6. $\epsilon$ is an algebra homomorphism, $\epsilon m=\epsilon \otimes \epsilon ,$
7. (antipode condition), $\mu \left(\mathrm{id}\otimes S\right)\Delta =\mu \left(S\otimes \mathrm{id}\right)\Delta =1\epsilon .$

In condition (5) the algebra structure on $A\otimes A$ is given by $$\left(a\otimes b\right)\left(c\otimes d\right)=ac\otimes bd,\text{for}a,b,c,d\in A,\text{and}\begin{array}{rcll}\tau :& \otimes A& \to & A\otimes A\\ & a_1\otimes a_2& \mapsto & a_2\otimes a_1\end{array}.$$ In condition (6) we have identified the vector space $𝕂\otimes 𝕂$ with $𝕂.$ Since $$\text{???}$$ the antipode $S:A\to A$ is an antihomomorphism $$S\left(a_1{a}_2\right)=S\left(a_2\right)S\left(a_1\right),\text{for all}{a}_1,a_2\in A.$$

Let $A$ be a Hopf algebra over $𝕂.$ If $a\in A$ 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 $A\otimes A.$ This notation is called Sweedler notation and is a standard notation for working with Hopf algebras. It shouldn't be bothersome, it is simply way to actually write $\Delta \left(a\right)$ so that it looks like an element of $A\otimes A$ without having to go through all the rigmarole of actually choosing a basis for $A.$

For $A$-modules $M,N$ and $P,$ define the tensor product to be the $A$-module $M\otimes N$ with $A$-action given by $$a\left(m\otimes n\right)=\Delta \left(a\right)\left(m\otimes n\right)=\sum _a{a}_{\left(1\right)}m\otimes a_{\left(2\right)}n,\text{if}\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)},$$the trivial module $\iota =𝕂.1$ with $𝕂$-action given by $$a.1=\epsilon \left(a\right),$$ and the dual module $M*={\mathrm{Hom}}_{𝕂}\left(M,𝕂\right)$ with $A$-action given by $$\left(a\phi \right)m=\phi \left(S\left(a\right)m\right),\text{for}\phi \in M*,m\in M.$$

The definition of a Hopf algebra is exactly designed so that $M\otimes N,{𝕂}_{\epsilon }$ and $M*$ are well defined $A$-modules and the maps $$\begin{array}{rcl}\left(M\otimes N\right)\otimes P& \to & M\otimes \left(N\otimes P\right)\\ \left(m\otimes n\right)\otimes p& \mapsto & m\otimes \left(n\otimes p\right)\end{array},$$$$\begin{array}{rcl}M\otimes \iota & \to & M\\ m\otimes \iota & \mapsto & m\end{array}\text{and}\begin{array}{rcl}\iota \otimes M& \to & M\\ \iota \otimes m& \mapsto & m\end{array}$$ and $$\begin{array}{rcl}M*\otimes M& \to & \iota \\ m\otimes \phi & \mapsto & \phi \left(m\right)\end{array}\text{and}\begin{array}{rcl}\iota & \to & M\otimes M*\\ 1& \mapsto & \sum _i{e}_i\otimes e^i\end{array}$$ are $A$-module homomoorphisms. The sum in (???) is over a $𝕂$-basis $\left\{e_i\right\}$ of $M$ and we only consider this map when this sum exists. CHECK on and REMARK on the order of $M$ and $M*$ in the tensor products.

Let $A$ be a Hopf algebra. The vector space $A$ is an $A$-module where the action of $A$ on $A$ is given by $$\begin{array}{rcl}A\otimes A& \to & A\\ a\otimes b& \mapsto & \sum _a{a}_{\left(1\right)}bS\left(a_{\left(2\right)}\right),\end{array}\text{where}\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}.$$

The linear transformation of $A$ determined by the action of an element $a\in A$ is denoted by ${\mathrm{ad}}_a.$ Thus $${\mathrm{ad}}_a\left(b\right)=\sum _a{a}_{\left(1\right)}bS\left(a_{\left(2\right)}\right),\text{for all}b\in A.$$

Let $M$ be an $A$-module and let $\rho :A\to \mathrm{End}\left(M\right)$ be the corresponding representation of $A$, ie the map $$\begin{array}{rcll}\rho :& A& \to & \mathrm{End}\left(M\right)\\ & a& \mapsto & \rho \left(a\right)\end{array}$$ where $\rho \left(a\right)$ is the linear transformation of $M$ determined by the action of $a.$ Note that $\mathrm{End}\left(M\right)\cong M\otimes M*$ as a vector space. On the other hand, $M\otimes M*$ is an $A$-module. The definition of the adjoint action is exactly designed so that the composite map $$\rho :A\to \mathrm{End}\left(M\right)\cong M\otimes M*\text{is an }A\text{-module homomorphism.}$$

Let $A$ be a Hopf algebra with antipode $S$ and let $M$ be an $A$-module. A bilinear form $$\begin{array}{rcll}⟨,⟩:& M\otimes M& \to & 𝒦\\ & m\otimes n& \mapsto & ⟨m,n⟩\end{array}\text{is <strong>invariant</strong> if}⟨a{m}_1,m_2⟩=⟨m_1,S\left(a\right)m_2⟩,$$ for all $a\in A,m_1,m_2\in M.$ This is equivalent to the conditiomn that the map $⟨,⟩$ is a homomorphism of $A$-modules when we identify $𝕂$ with the trivial $A$-module $\iota .$

A bilinear form $$⟨,⟩:A\otimes A\to 𝕂\text{is <strong>ad-invariant</strong> if}⟨{\mathrm{ad}}_a\left(b_1\right),b_2⟩=⟨b_1,{\mathrm{ad}}_{S\left(a\right)}\left(b_2\right)⟩,$$ for all $b_1,b_2\in A.$ In other words, the bilinear form is invariant if we view $A$ as an $A$-module via the adjoint action.

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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