Hopf algebras

Our base ring R will always be assumed commutative, and whenever ( )⁰ appears we happily suppose it to be a field.

An R- Hopf algebra is an R-bialgebra H together with R-module morphism

$$\map \nu \from H \to {H}$$

called the antipode, which satisfies the following diagram.

$$\spreaddiagramcolumns {-2pc}\diagram H \rto^-{\delta}\drrto... ...\ \hskip 6pc & & R \urrto_-{\;\eta} & & \hskip 6pc \enddiagram$$

For any Hopf algebra H let $H^\op$ denote the Hopf algebra obtained by replacing $\mu$ with $\mu\circ\sigma$ : $\mapnamed{\,\sigma}\from H \XO H \to {\mapnamed \mu \from H \XO H \to H}$ and replacing $\delta$ with $\delta\circ\sigma$ : $\mapnamed{\,\delta}\from H \to {\mapnamed \sigma \from H \XO H \to H \XO H}$ while keeping the same $\eta$, $\varepsilon$ and $\nu$.

There is also a bialgebra $H^{\prime}$ obtained more simply by just replacing $\delta$ with $\delta\circ\sigma$ : $\mapnamed \delta \from H \to {\mapnamed \sigma \from H \XO H \to H \XO H}$ while keeping the same $\mu$, $\eta$, $\varepsilon$ and $\nu$. In general however, this $H^{\prime}$ is not a Hopf algebra.

Proposition 10.1  

Let H be a Hopf algebra. Then

(a) the antipode $\nu$ is uniquely determined;

(b) $\map \nu \from H^\op \to H$ is a bialgebra morphism;

(c) $H^{\prime}$ is a Hopf algebra if and only if $\nu$ is bijective (moreover the antipode for $H^{\prime}$ is the inverse for $\nu$);

(d) if H is commutative or cocommutative then $\nu \circ \nu = 1_H$ (that is, $\nu$ is an involution).

Proposition 10.2  

Let H and K be any Hopf algebras. Then each bialgebra morphism $\map f \from H \to K$ preserves antipode.

$$\spreaddiagramcolumns{1pc}\spreaddiagramrows{.5pc}\diagram H... ...\rto^-{f\;} & K \dto^-{\,\nu} \\ H \rto^-{f\;} & K \enddiagram$$

Using other fancier words, the category $\Hopf_R$ of Hopf algebras is a full subcategory of the category $\BIG_R$ of bialgebras.

For any algebra H we have seen that H⁰ becomes a coalgebra. If H is a bialgebra then H⁰ becomes a bialgebra using the multiplication

$$H^0_{}\XO H^0_{} \;\isom\; \longmapnamed<2pc>{\;\delta^0_{}}\from (H \XO H)^0_{} \to {H^0_{}}$$

and unit

$$R \;\isom\; \longmapnamed<2pc>{\;\varepsilon^0_{}}\from R^0_{} \to {H^0_{}}$$

(recall the first proposition in the previous section). Furthermore, if H is a Hopf algebra then so is H⁰ with antipode

$$\map \nu^0_{} \from \,H^0_{}\, \to {\,H^0_{}}\,.$$

What we have here is a contravariant ``self-adjoint'' functor

$$\map (\slot)^0_{} \from \,\Hopf_R^{\;\op}\, \to {\,\Hopf_R}\;.$$

What ``self-adjoint" means in this context is that

$$\BIG_R(H,K^0_{}\,)\;\;\isom\;\;\BIG_R(K\,,H^0_{}\,) \;.$$

Proposition 10.3  

If H is any Hopf algebra then the monoid $\scrD(H)$ of set-like elements is a group.

An A-point of a Hopf algebra H is an algebra morphism $\map f \from H \to A$.

Proposition 10.4  

(a)If $\map f\,,g \from H \to A$ are commuting A-points of H (meaning that [ f(h) , g(k) ] = 0 for all $h\,,k \in H\,$) then $\map f*g \from H \to A$ is an A-point of H.

(b)If $\map f \from H \to A$ is an A-point of H then f has a convolution inverse $\map f \circ \nu \from H^\op \to A$ which is an A-point of $H^\op$.

Example 10.5  

For a monoid G, we have seen that the monoid algebra R(G) is a bialgebra. If G is a group then the group algebra R(G) becomes a Hopf algebra with antipode $\map \nu \from R(G) \to {R(G)}$ given by $\nu(g)=g^\inv$. (The axioms for $\map (\slot)^\inv \from G \to G$ expressed diagramatically in $\Set$ are taken by the functor $\map \Free_R \from \Set \to {\Mod_R}$ into the axioms which define the antipode.)

Example 10.6  

For a Lie algebra L, write $L^\op$ for the Lie algebra with the same module L but with Lie bracket $\beta^\op$ given by $\beta^\op(x\,,y\,)=\beta(\,y\,,x\,)$. For any algebra A we have $(A^\op)_L^{}=(A_L^{})^\op$. It follows (why?) that we have a canonical algebra isomorphism

$$\Univ\!(L^\op) \;\;\isom\;\; \Univ\!(L)^\op \;.$$

We have a Lie algebra isomorphism $\map\from L \to {L^\op}$ taking x to -x (note that [ -x ,-y ]=[ x ,y ]=-[ y,x ] ). So we define $\map \nu \from \Univ\!(L) \to {\Univ\!(L)^\op}$ by

$$\spreaddiagramcolumns{1pc}\spreaddiagramrows{.5pc}\diagram L... ...v\!(L)^\op \;\rlap{$\;\isom\;\Univ\!(L^\op_{})\;.$} \enddiagram$$

One easily checks that for $x_1^{}\,,\,\ldots\,, \,x_n^{} \in L$

$$\nu(i(x_1^{})\, \cdots\, i(x_n^{})) \;\;=\;\; (-1)^n \,i(x_1^{})\,\cdots\,i(x_n^{}) \;.$$

With this antipode $\Univ\!(L)$ becomes a Hopf algebra.

Example 10.7  

The matrix bialgebra M!(n) (earlier example ) is not a Hopf algebra. We need to ``adjoin an inverse for the determinant''. Recall that $M\!(n) = R[\,X\,] = \Symm\big(\Free_R(X)\big)$ where $X=\{x_{ij}^{}\,\vert\;i\,,j=\,1,\,\dots\,,n\}$ has cardinality n². Define

$$\det(X)\;\;=\;\;\sum_{\xi\in \Symm_{\!n}^{}} (-1)^{\vert\xi\... ...x_{1\,\xi(1)}^{}\, x_{2\,\xi(2)}^{} \,\dots\, x_{n\,\xi(n)}^{}$$

where $\vert\xi\vert$ is the least number of simple transpositions required to obtain the permutation $\xi$. Form the following commutative polynomial R-algebra: $R[\,X\cup\{t\}\,]\;=\;R[\,(x_{ij}^{})\,,t\,]\; =\;\Symm\big(\Free_R(X\cup \{t\})\big)$, in n²+1 (commuting) indeterminates t and x_ij with $(1\le i\,,j\le n)$. Put

$$\GL(n)\;\;=\;\; R[\,X\cup\{t\}\,]/(\,t\det(X)-1\,)$$

as a commutative R-algebra. We make $\GL(n)$ into a bialgebra by defining

$$\vcenter{ \halign{\hfil$ ...$$

modulo $(\,t\det(X)-1\,)$. Put $X_{ij}^{}=\{x_{rs}^{}\,\vert\;r\ne i\,,\,s\ne j\,\}$. Now define the morphism $\map \nu \from \GL(n) \to {\GL(n)}$ by

$$\begin{eqnarray*} \nu(x_{ij}^{}) &=& t\,\det(X_{ji}^{}\,) \\ \nu(t)\;\; &=& \det(X) \end{eqnarray*}$$

modulo $(\,t\det(X)-1)$. Then $\GL(n)$ becomes a Hopf algebra.

For any commutative R-algebra A we have a canonical isomorphism of groups

$$\Alg_R(\GL(n)\,, A) \;\;\isom\;\; \GL(n\,,A) \;.$$

The first and second examples above exhibit cocommutative Hopf algebras R(G) and $\Univ\!(L)$, while the previous example is a commutative Hopf algebra $\GL(n)$. It is only recently that the importance of Hopf algebras which are neither commutative nor cocommutative has been properly understood.

Example 10.8  

We now describe a ``quantum deformation'' of the previous example. This is a generalization to n x n, from the 2 x 2 case discussed in an earlier section.

Take $X=\{x_{ij}^{}\,\vert\, i\,,j=1\,,\,\dots\,,n\}$ as in the previous example. First we form the free algebra $R\langle X\,\rangle=T\big(\Free_R(X)\big)$ on the (non-commuting) indeterminates x_ij. Let $\Mq(n)$ denote the quotient of $R\langle X\,\rangle$ be the ideal generated by the following elements:

$$\begin{eqnarray*} & x_{ir}^{}\, x_{jk}^{}-x_{jk}^{}\, x_{ir}^{}\qquad &\hbox{\e... ...r}^{}-q\,x_{ir}^{}\, x_{ik}^{}\qquad & \hbox{\em for $\;k<r$~.} \end{eqnarray*}$$

This becomes a coalgebra with comultiplication

$$\delta(x_{ij}^{})\;\;=\;\;\sum_{r=1}^n x_{ir}^{}\xO x_{rj}^{} \quad\rlap{\rm ( modulo the ideal )}$$

and counit

$$\varepsilon (x_{ij}^{})\;\;=\;\;\delta_{ij}^{} \qquad\rlap{\rm ( Kronecker delta )~.}$$

Define the ``quantum determinant'' by

$$\detq(X) \;\;=\;\; \sum_{\xi\in \Symm_{\!n}^{}} (-q)^{\vert\... ...,x_{1\,\xi(1)}^{}\,x_{2\,\xi(2)}^{}\,\cdots\, x_{n\,\xi(n)}^{}$$

which is a central element of $\Mq(n)$ (that is, it commutes with all other elements). The quantum general linear group is defined by

$$\GL_{\,q}(n)\;\;=\;\;\Mq(n)[\,t\,]/(\,t\,\detq(X)-1\,) \;.$$

We adjust the comultiplication and counit of$\Mq(n)$ by defining $\delta(t) = t\xO t$ and $\varepsilon(t)=1$. Then we have a bialgebra epimorphism

$${\map \rho \from \Mq(n) \to {\GL_{\,q}(n)}}\;.$$

Define $\map \nu \from \GL_{\,q}(n) \to {\GL_{\,q}(n)}$ by

$$\nu(x_{ij}^{}) \;\;=\;\; t\,\detq(X_{ji}^{})\qquad\hbox{and}\qquad \nu(t) \;\;=\;\; \detq(X)\;.$$

Then $\GL_{\,q}(n)$ becomes a Hopf algebra. Notice that $\GL_{\,q}(n)^\op=\GL_{\,q^\inv}\!(n)$.

Many claims have been made in this section. For n=2 the calculations in an earlier theorem prove them all. (This should be compared with the last proposition.) The general case can be verified similarly, but will follow from later work.