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\bogustitlepage{10}{``Representations of quantum groups''}

\chapter{Representations of quantum groups}

\noindent
We mention in chapter 6, example 3 that a representation
of a group $G$ was an $R(G)$-module. One kind of
representation for a Hopf algebra $H$ therefore suggests
itself: a module over $H$. We begin by discussing modules
over bialgebras.

First note that if \map f \from E \to A is a ring morphism 
then each (left) $A$-module $M$ becomes a (left) $E$-module 
via the action
$$
em=f(e)m \qquad \hbox{for }e\in E \;, \ m \in M \;.
$$
This is called {\em restriction of scalars along $f$\/}.
\index{restriction of scalars}

Let $A$ be an $R$-algebra. Then each module is
automatically an $R$-module via restriction of scalars
along the unit \map \eta \from R \to {A}. Alternatively,
we can view an $A$-module as an $R$-module $M$ with a
ring morphism \map \hat \mu \from A \to {\End_R(M)}.
Later, we want to look at ``comodules'', 
\index{comodule}
and so we want a definition of $A$-module which dualizes. 
The good version is: an $R$-module $M$ with a module morphism
\map \mu \from A \xR M \to {M}, 
called the {\em action\/}\index{action} of $A$ on $M$, satisfying
\[
\diagram
A\xR A\xR M\rto<.5ex>^-{\smu\xo1\;}
											\rto<-.5ex>_-{1\xo\smu\;} & A\xR M \rto^-{\mu} & M \\
\enddiagram 
\]
\smallskip
\[
\diagram
& A \xR M \drto^-{\; \mu} & \\
M \urto^-{\eta \xo 1 \;} \rrto^{1} & & M \;. \\
\enddiagram
\]
We write $\Mod_R(A)$ for $\Mod_A$ just to
emphasize that we build it up from $\Mod_R$.

Suppose $M$ and $N$ are (left) modules over the $R$-algebra
$A$. Regard \modmap M \from A \to {R} and \modmap N \from R \to 
{A^{\op}}. We see that \modmap M \xR N \from A \to {A^{\op}}, 
which means $M \xR N$ becomes an $A \XO A$-module.
If $A$ is a bialgebra then we can restrict
scalars along \map \delta \from A \to {A \xR A} to
obtain an $A$-module structure on $M \xR N$.
Explicitly, the action is the composite
\[
\xymatrixcolsep{1.75pc}
\diagram
A \xR M \xR N \rto^-{\delta \xo 1 \xo 1 \;} & A \xR A \xR M \xR N
\rto^-{1 \xo \sigma\xo1 \;} & A \xR M \xR A \xR N 
\rto^-{\smu \xo\mu \;} & M\xR N\,.
\enddiagram
\]
This generalizes to multiple tensor products (over $R$) of $A$-modules. 
In particular, the empty tensor product $R$ becomes an $A$-module 
by restricting scalars along the counit \map\varepsilon\from A\to{R}.

With $M$ and $N\,$ left $A$-modules as before, we can regard
\modmap M\from R\to{A^{\op}} and 
\modmap N\from R\to{A^{\op}}, so that 
\modmap \Hom_R(M,N) \from A^{\op} \to {A^{\op}}; or in other words
$\Hom_R(M,N)$ becomes an $A^{\op}\XO A$-module. 
Thus if $A=H$ is a Hopf algebra, 
we can restrict scalars along the $R$-algebra morphism 
\[
\xymatrixcolsep{2pc}
\diagram
H \rto^-{\delta\;} & H\xR H \rto^-{\nu\xo1\;} & H^{\op}\xR H \\
\enddiagram
\]
to make $\Hom_R(M,N)$ into an $H$-module.
Explicitly, the action of $H$ on $\Hom_R(M,N)$ is
the composite
\[
\xymatrixcolsep{1.5pc}
\xymatrixrowsep{0pc}
\diagram
H\xR \Hom_R(M,\!N)\rto^-{\delta\xo1\;} &
H\xR H\xR\Hom_R(M,\!N)\arrow [r]^-(.7){1\xo\sigma\;}&\\
H\xR\Hom_R(M,\!N)\xR H\rto^-{\smu_1^{}\xo\nu\;} &
\Hom_R(M,\!N)\xR H \rto^-{\smu_2^{}} & \Hom_R(M,\!N) \\
\enddiagram
\]
where $\mu_1^{}$ and $\mu_2^{}$ are the left and right actions \dots
\[
\def\smapsto{\morphism{\solid}{\tip}{\stop}[0,2]}
\def\lmapsto{\morphism{\solid}{\tip}{\stop}[0,4]}
\def\lline{\morphism{\solid}{\tip}{}[0,3]}
\xymatrixcolsep{-2pc}
\spreaddiagramrows{-2pc}\diagram
H\xR \Hom_R(M,\!N)\rrto^-{\smash{\hat\mu}\xo1\;} &&
 \Hom_R(N,\!N)\xR\Hom_R(M,\!N)\arrow[rr]^-{\circ\;} 
 &\hskip5pc & \hskip1.4pc & \Hom_R(M,\!N) \\
\mu_1^{}\;:\; h\xo f\;\arrow @{|->}[rrr] &\hskip5pc &
&&\span{(m\;\mapsto h(fm))}\\
\Hom_R(M,\!N) \xR H \rrto^-{1\xo\smash{\hat\mu}\;} &&
 \Hom_R(M,\!N)\xR\Hom_R(M,\!M)\arrow[rrr]^-{\circ\;} 
 & \hskip 5pc & \hskip 1.4pc & \Hom_R(M,\!N) \\
\mu_2^{}\;:\; f\xo h\;\arrow @{|->}[rrr] &\hskip5pc &
&&\span{(m\;\mapsto f(hm))}\rlap{~.}\\
\enddiagram
\]

\goodbreak

\begin{proposition}
For left modules $M$ and $N$ over the Hopf algebra $H$, 
the canonical $R$-module morphisms
\begin{eqnarray*}
{\map e \from \Hom_R(M,N) \xR M \to {N}} &\hbox{\rm where }
 & f\xo m\;\mapsto\; f(m)\\
{\map d \from M \to {\Hom_R(N,M \xR N)}}&\hbox{\rm where } 
 & m\; \mapsto\; (n\;\mapsto\; m\xo n)
\end{eqnarray*}
are left $H$-module morphisms.
\end{proposition}

\begin{followon}{Proof}
Omitting $\xR$ and $\Hom_R$ from the notation, we
obtain the first of these from the following diagram. 
The second we leave to the reader.
\[
\xymatrixcolsep{1.5pc}
\diagram
H(MN)M \ddto_-1\rto^-{\delta11\;} 
 & HH(MN)M\dto< .5ex>_-{1\delta11\;\;}
          \dto<-.5ex>^-{\;\;\delta111} 
  \arrow @/_10pt/[ddl]_-{1\varepsilon11}\rto^-{1\sigma1}
 & H(MN)HM \drto^-{\quad\delta111} &\\
& HHH(MN)M\dto^{1\nu11}\rrto^-{11\sigma1\;}&& HH(MN)HM \dto^-{1\sigma11}\\ 
H(MN)M \arrow [dr]^-{1\eta11}\ddrto_-{11\eta1}
 \arrow @/_20pt/[dddr]_-{1} 
 & HHH(MN)M \dto^-{1\mu11}\rto^-{11\sigma1\;} 
 & HH(MN)HM \drto^-{1\sigma11}& H(MN)HHM \dto^-{1\nu11}\\
& HH(MN)M \dto^-{1\sigma1} && H(MN)HHM \dllto_-{11\mu1}\dto^-{1\mu_2 11}\\
& H(MN)HM \dto^-{1\mu_2 1} && H(MN)HM \dllto_-{1\mu_2 1}\dto^-{\mu_1 11}\\
& H(MN)M \dto^{1e}\rto_-{\smu_1^{}1\;}&(MN)M\dto^-{e}&(MN)HM\lto_-{\;1\mu}\\
& HN \rto^-{\smu\;} & N \\  
\enddiagram
\]
\end{followon}

\begin{followon}{Corollary 2} For modules $M$, $N$ and $L$ over a
Hopf algebra $H$, the canonical isomorphism
\[
\Hom_R(M\xR N\,,L) \;\;\isom\;\; \Hom_R(M\,,\Hom_R(N,L))
\]
restricts to an isomorphism
\[
\Hom_H(M\xR N\,,L) \;\;\isom\;\; \Hom_H(M\,,\Hom_R(N,L)) \;.
\]
\end{followon}

\begin{followon}{Proof} The canonical isomorphism is obtained from
the evaluation $e$ and the canonical $d$ of Proposition 1.
\end{followon}

\noindent
In other words, we have a nice {\em tensor}--$\hom$ situation for the
category $\Mod_R(H)$ of (left) $H$-modules. 
Both the tensor and the $\hom$ are preserved by the functor
\[
\map \from \Mod_R(H) \to \Mod_R
\]
given by ignoring the $H$-action.

Although modules over the group algebra are representations of the
group, so that the study of modules over a Hopf algebra does
suggest itself, the point of view of Section 2 
(i.e. space--algebra duality) 
leads more naturally to ``comodules''. For here, it is the
comultiplication \map \delta \from H \to {H\XO H} of the Hopf
algebra which corresponds to the spatial multiplication.

Suppose $C$ is an $R$-coalgebra. A (left) $C$-{\em comodule\/} 
\index{comodule, left}
is an $R$-module M with a module morphism 
\map \delta \from M \to {C\xR M}, 
called the {\em coaction\/}\index{coaction } 
of $C$ on $M$, satisfying
\[
\xymatrixcolsep{2.5pc}
\diagram
M \rto^-{\delta\;} & C\xR M \rto<.5ex>^-{\delta\xo1\;}
													            \rto<-.5ex>_-{1\xo\delta\;} 
& C\xR C\xR M \\
\enddiagram 
\]
\smallskip
\[
\diagram
& C\xR M \drto^-{\varepsilon\xo1} & \\
 M \urto^-{\delta}\rrto^-{1\;} && M \\
\enddiagram
\]
We write $\Com_R(C)$ for the category whose objects are
$C$-comodules and whose arrows are $C$-{\em comodule morphisms\/};
\index{comodule morphism}\index{morphism \subitem comodule }
that is, $R$-module morphisms \map f\from M\to N such that
\[
\diagram
M \dto_{f} \rto^-{\delta\;} & C\xR M \dto^-{1\xo f} \\
N \rto^-{\delta\;} & C\xR N \\
\enddiagram
\]

Each $C$-comodule $M$ becomes a $C^*$-module with the action 
\[
\diagram
C^*\xR M \rto^-{1\xo\delta\;} & C^*\xR C\xR M 
 \rto^-{e\xo1\;} & M \;. \\
\enddiagram
\]
By the fundamental theorem of Morita theory (Theorem 5.2),
\index{Morita Theory }
if $C$ is cauchy (as an $R$-module) then this gives a bijection between
$C$-coactions $\delta$ and $C^*$-actions $\mu$ on each $R$-module $M$: 
recover $\delta$ as \dots
\[
\diagram
M \rto^-{d\xo1\;} & C\xR C^* \xR M \rto^-{1\xo\smu\;} & C\xR M\;. \\
\enddiagram
\]
So for $C$ cauchy, we have an isomorphism of categories
\[
\Com_R(C)\;\;\isom\;\;\Mod_R(C^*)\;.
\]

If $C$ is an $R$-bialgebra not necessarily cauchy we obtain, in
a manner dual to that for modules, a coaction on the tensor product
(over $R$) of $C$-comodules. Explicitly for $C$-modules $M$ and $N$,
the coaction for $M\xR N$ is given by the composite
\[
\xymatrixcolsep{1.85pc}
\diagram
M \xR N \rto^-{\delta\xo\delta\,} & C\xR M\xR C\xR N
\rto^-{\sigma_{1324}\,} & C\xR C\xR M\xR N \rto^-{\smu\xo1\xo1\,}
 & C\xR M\xR N\rlap{~.}
\enddiagram
\]
The empty tensor product $R$ has the coaction 
\map\eta\from R\to{C\xR R}.

When it comes to $\Hom$ our formal duality fails: in reversing
arrows we have maintained $\XO$, yet $\Hom$ does not maintain its
universal property.
However if $M$ is cauchy, $\Hom_R(M,N)$ does have the
{\em reverse-arrow universal property\/}:
\index{reverse-arrow universal property}
\index{universal property \subitem reverse-arrow }
there is a bijection between $R$-module morphisms
\[
\diagram L & \Hom_R(M,N) \lto \enddiagram
\]
and $R$-module morphisms
\[
\diagram M\XO L & N \lto \enddiagram
\]
since $\Hom_R(M,N)\isom M^*\xR N$ and $M\xR L\isom\Hom_R(M^*,L)$.

\begin{proposition} Each cauchy $R$-module M gives rise
to an $R$-coalgebra $M \xR M^*$ with counit 
\map e \from M \xR M^* \to R and comultiplication
\[
\map 1\xo d\xo 1 \from M\xR M^* \to {M\xR M^*\xR M\xR M^*}
\]
(see Theorem 5.2). For any $R$-coalgebra $C$, the assignment
\[
\diagram
 \delhat\;=\; (M \xR M^* \rto^-{\delta\xo1\;} & C \xR M\xR M^*
 \rto^-{1\xo e\;} & C) \\
\enddiagram  
\]
determines a bijection between coactions
\[
 \map\delta\from M\to{C\xR M}
\]
of $C$ on $M$ and coalgebra morphisms
\[
 \map\delhat\from M\xR M^*\to{C}.
\]
\end{proposition}

\begin{followon}{Proof} 
$M \xR M^* \isom_{\sigma} M^*\xR M \isom_{\rho}\Hom_R(M,M)$ 
has the universal property of $\Hom$ under arrow reversal; 
so the diagrammatic proof that $\End_R(M)$ is an algebra and that 
an action is an algebra morphism \map\from A\to{\End_R(M)}, dualizes.
\end{followon}

\noindent
Take $M = R^n$ in the above proposition and let $e_1^{},\ldots,e_n^{}$
be the standard basis. 
Now let $e_1^*, \ldots, e_n^*$ be the dual basis for ${R^{n}_{}}^*$; 
that is, $e_i^*(e_j^{}) = \delta_{ij}^{}$ (Kronecker $\delta$). 
A coaction of $C$ on $R^n$ thus amounts to a coalgebra
morphism \map \hat \delta \from R^n_{} \xR {R^{n}_{}}^*\to {C}, 
and this is determined by its values on the basis elements 
$e_i^{}\xo e_j^*$ of $R^n_{}\xR {R^{n}_{}}^*$:
\[
 \delhat(e_i^{}\xo e_j^*) = x_{ij}^{} \in C \;.
\]
So $C$-comodule structures on $R^n_{}$ are in bijection 
with {\em multiplicative matrices\/}%
\index{multiplicative matrices}
in $C$; that is, matrices $\xx = (x_{ij}^{})$ in $C$ satisfying
\[
\delta(x_{ij}^{}) = \sum_k x_{ik}^{}\xo x_{kj}^{}\;, \qquad
\varepsilon(x_{ij}^{}) = \delta_{ij}^{} \;. 
\]
Following Manin, we write the last two equations as
\[
\delta(\xx) = \xx\xo\xx \;, \qquad \varepsilon(\xx) = \ii\;,
\]
where $\ii$ is the identity matrix and 
$\xx\xo\yy = (\sum_k x_{ik}^{}\xo x_{kj}^{})$ 
is {\em not\/} the usual tensor product of matrices.

\begin{example} 
In the situtation of Example 8.4, 
$\xx = (x_{ij}^{})$ and $\pmatrix{\xx&0\cr 0&t\cr}$ are
multiplicative matrices for $\Mat_q(n)$ and $\GL_{\,q}(n)$, respectively.
\end{example}

Now suppose $C=H$ is a Hopf algebra and $M$ is a cauchy $R$-module.
By applying Proposition 3 to $M^*_{}$ 
(and using the canonical $M^{**}_{}\isom M$), 
we see that $M^*_{}\xR M$ becomes a coalgebra with counit
\[
\diagram
M^*\xR M \rto^-{\sigma\;} & M\xR M^* \rto^-{e\;} & R \\
\enddiagram
\]
and comultiplication
\[
\spreaddiagramcolumns{1pc}
\diagram
M^* \xR M \rto^-{1\xo d\xo1 \;} 
& M^*\xR M^*\xR M \xR M \rto^-{1\xo\sigma\xo1\;} 
&  M^* \xR M\xR M^*\xR M \;. \\
\enddiagram
\]

\begin{proposition} 
(a) \map \delhat \from M\xR M^* \to H 
is a coalgebra morphism iff the composite
\[
\diagram
M^*\xR M \rto^-{\sigma\;} & M\xR M^* \rto^-{\delhat\;} & H^{\op} \\
\enddiagram
\]
is a coalgebra morphism.

\noindent (b) Suppose $M$ is an $H$-comodule and put
\[
\diagram
\delhat_k = (M\xR M^*\rto^-{\delhat\;} & H\rto^-{\nu^k_{}\;} & H\;.\\ 
\enddiagram 
\]
For $k$ even, $\delhat_k$ is a coalgebra morphism and has
convolution inverse $\delhat_{k+1}^{}$ in $\Hom_R(M\xR M^*,H)$.
\end{proposition}

\begin{followon}{Proof} (a) 
\[
\diagram
M^*M \dto_-{1\xo d\xo1} \rto^-{\sigma\;} & MM^* \dto^-{1\xo d\xo1}
 \rto^-{\delhat\;} & H \dto^-{\delta} \\
M^*M^*MM \dto_-{1\xo\sigma\xo1}\rto^-{\sigma_{4231}\;} &
 MM^*MM^* \dto^-{\sigma_{3412}}\rto^-{\delhat\xo\delhat\,} 
 & H\XO H \dto^-{\sigma} \\
M^*MM^*M \rto^-{\sigma\xo\sigma\;}& MM^*MM^*\rto^-{\delhat\xo\delhat\,} 
 & H \XO H \\
\enddiagram
\quad
\xymatrixcolsep{1.25pc}
\diagram
M^*M \rto^-{\sigma\;} \dto_-{\sigma} 
 & MM^* \ddto^{e} \rto^-{\delhat} \arrow @{=}[dl] 
 & H \ddlto^-{\varepsilon} \\
MM^* \drto^-{e\;}\\  & R \\
\enddiagram
\]

\noindent (b) Since \map \nu\from H\to{H^{\op}} is a coalgebra
morphism, by Proposition 9.1(b); 
then \map \nu^k_{}\from H\to H is a coalgebra morphism for $k$ even; 
so $\delhat_k^{} = \nu^k_{}\circ\delhat$ is a coalgebra morphism, as required. 
This also means
that right composition with $\delhat_k$ preserves convolution.
Since $1_H^{}$ and $\nu$ are convolution inverses, 
so are $1_H^{}\circ \delhat_k^{}$ and $\nu\circ\delhat_k^{}$; 
that is, so are $\delhat_k^{}$ and $\delhat_{k+1}^{}$.
\end{followon}

\begin{proposition} Suppose that $M$ is a comodule over
the Hopf algebra $H$ and that $M$ is cauchy as an $R$-module. 
Then $M^*$ becomes an $H$-comodule via
\[
\diagram
\delhat = (M^*\xR M \rto^-{\sigma\;} & M\xR M^*\rto^-{\delhat\;} 
 & H \rto^-{\nu\;} & H)\;.
\enddiagram
\]
Moreover, the $R$-module morphisms
\[
\displaylines{
{\map e \from M \xR M^*\to R} \cr
{\map d \from R \to {M^*\xR M}} \cr}
\]
become $H$-comodule morphisms. 
One might therefore say that $M$ becomes a {\em cauchy $H$-comodule}.
\index{cauchy $H$-comodule}
\end{proposition}

\begin{followon}{Proof} 
By Propositions 10.4(a) and 9.1(b), the stated $\delhat$ 
is a coalgebra morphism and so, by Proposition~10.3, 
determines a coaction of $H$ on $M^*$. 
Tracing through, one sees that this is dual to the situation 
for $\Hom_R(M,R)$ as in Proposition 10.1; 
so the proof dualizes, but it can also be shown
directly that $e$ and $d$ are comodule morphisms.
\end{followon}

\noindent {\bf Remark:} To obtain results as in Proposition~10.4(b)
for $k$ odd, apply Proposition~10.4(b) to the $M^*$ of 
Proposition~10.5; compare with \cite{Manin}, p.14.

\clearpage
\bogusendpage