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Real structures

The usual approach to real structures is to consider $\bC$ -Hopf algebras endowed with a *-structure.
$\begin{defn} A Hopf-*-algebra is a Hopf algebra over $\bC$\ endowed with the un... ...orphism $*\:A\to A$ such that $\Dl$\ and $\eps$\ are *-homomorphisms. \end{defn}$
One can then prove that $*\circ S = S^{-1}\circ *$ .
$\begin{prop} Let $G$\ be a complex algebraic group with Lie algebra $\gerg$. The... ...ctures on $U(\gerg)$ \item Hopf-* structures on $F[G]$ \end{enumerate}\end{prop}$

$\begin{defn} A \textbf{real quantum group} is a global quantized function algebra with a compatible *-structure. \end{defn}$

$\begin{example} Consider the example of $F_q[\GL_n(\bC)]$. Fix on it the *-stru... ...led the unitary $F_q[\rU(n)]$ (compact form of $F_q[\GL_n(\bC)]$). \end{example}$

$\begin{example} Let $0<q<1$. Consider the *-algebra generated by $\al, \ga$ ($=t... ...*}This is called the (standard) quantum $\SU_q(2)$; $F_q[\SU(2)]$. \end{example}$

$\begin{example} Let $0<q<1$. Consider the *-algebra generated by $v, n$\ subject... ...lign*}This is called the (standard) quantum $E_q(2)$; $F_q[E(2)]$. \end{example}$

$\begin{example} Consider now the *-algebra generated by $v, n$\ subject to relat... ...cture as before. This is called the non standard quantum $E_q(2)$. \end{example}$

Pawel Witkowski 2006-06-26