Hopf algebra HopfAlgebra 2002-10-18 14:10:26 2005-07-12 00:05:33 Definition antipode \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % making logically defined graphics \usepackage[all]{xy} % my maths package \newcommand*{\Nset}{\mathbb{N}} \newcommand*{\Zset}{\mathbb{Z}} \newcommand*{\Qset}{\mathbb{Q}} \newcommand*{\Rset}{\mathbb{R}} \newcommand*{\Cset}{\mathbb{C}} \newcommand*{\Hset}{\mathbb{H}} \newcommand*{\Oset}{\mathbb{O}} \newcommand*{\Bset}{\mathbb{B}} \newcommand*{\Kset}{\mathbb{K}} \newcommand*{\Sset}{\mathbb{S}} \newcommand*{\Tset}{\mathbb{T}} \newcommand*{\GLgrp}{\mathrm{GL}} \newcommand*{\SLgrp}{\mathrm{SL}} \newcommand*{\Ogrp}{\mathrm{O}} \newcommand*{\SOgrp}{\mathrm{SO}} \newcommand*{\Ugrp}{\mathrm{U}} \newcommand*{\SUgrp}{\mathrm{SU}} \newcommand*{\e}{\mathop{\mathrm{e}}\nolimits} \newcommand*{\im}{\mathord{\mathrm{i}}} \newcommand*{\identity}{\mathord{\mathrm{1\!\!\!\:I}}} \newcommand*{\tr}{\mathop{\mathrm{tr}}} \newcommand*{\Tr}{\mathop{\mathrm{Tr}}} \renewcommand*{\d}{\mathrm{d}} \newcommand*{\deriv}[2]{\frac{\d #1}{\d #2}} \newcommand*{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand*{\fderiv}[2]{\frac{\delta #1}{\delta #2}} % my noncommutative geometry package \newcommand*{\algebra}[1][A]{\mathord{\mathcal{#1}}} \newcommand*{\hilbert}[1][H]{\mathord{\mathcal{#1}}} \newcommand*{\hilbmod}[1][E]{\mathord{\mathcal{#1}}} \newcommand*{\Matrix}[2]{\mathord{\mathrm{M}_{#1}(#2)}} \newcommand*{\dixmier}{\mathop{\mathrm{Tr}_\omega}} \newcommand*{\Res}{\mathop{\mathrm{Res}}} \newcommand*{\Wres}{\mathop{\mathrm{Wres}}} \newcommand*{\Aut}{\mathop{\mathrm{Aut}}\nolimits} \newcommand*{\Inn}{\mathop{\mathrm{Inn}}\nolimits} \newcommand*{\Out}{\mathop{\mathrm{Out}}\nolimits} \newcommand*{\Diff}{\mathop{\mathrm{Diff}}\nolimits} \newcommand*{\Ker}{\mathop{\mathrm{Ker}}\nolimits} \newcommand*{\Coker}{\mathop{\mathrm{Coker}}\nolimits} \newcommand*{\Img}{\mathop{\mathrm{Im}}\nolimits} \newcommand*{\End}{\mathop{\mathrm{End}}\nolimits} \newcommand*{\spin}{\mathop{\mathrm{spin}}\nolimits} \newcommand*{\Ind}{\mathop{\mathrm{Ind}}\nolimits} \newcommand*{\KK}{\mathit{KK}} \newcommand*{\HH}{\mathit{HH}} \newcommand*{\HC}{\mathit{HC}} \newcommand*{\ch}{\mathop{\mathrm{ch}}\nolimits} % my category theory package \newcommand*{\mathcat}[1]{\mathord{\mathbf{#1}}} \newcommand*{\id}{\mathrm{id}} \newcommand*{\op}{\mathrm{op}} \newcommand*{\boxprod}{\mathbin{\square}} % my environments \newtheoremstyle{inlinedefn}{}{0pt}{}{}{\bfseries}{.}{0.5em}{} \theoremstyle{inlinedefn} \newtheorem{definition}{Definition} \newtheoremstyle{break}{\baselineskip}{\baselineskip}{\itshape}{}{\bfseries}{}{\newline}{} \theoremstyle{break} \newtheorem{example}{Example} % misc commands \newcommand*{\defn}[1]{\textbf{#1}} A \textbf{Hopf algebra} is a bialgebra $A$ over a field $\Kset$ with a $\Kset$-linear map $S : A \to A$, called the \textbf{antipode}, such that \begin{equation} m\circ(S\otimes\id)\circ\Delta = \eta\circ\varepsilon = m\circ(\id\otimes S)\circ\Delta, \end{equation} where $m : A\otimes A \to A$ is the multiplication map $m(a\otimes b) = ab$ and $\eta : \Kset \to A$ is the unit map $\eta(k) = k\identity$. In \PMlinkescapetext{terms} of a commutative diagram: \[\begin{xy} \xymatrix@R=20pt@C=70pt{ & *+<10pt>\txt{$A$} \ar_{\Delta}[dl] \ar_{\varepsilon}[dd] \ar^{\Delta}[dr] & \\ *+<10pt>\txt{$A\otimes A$} \ar_{S\otimes\id}[dd] & & *+<10pt>\txt{$A\otimes A$} \ar^{\id\otimes S}[dd] \\ & *+<10pt>\txt{$\Kset$} \ar_{\eta}[dd] & \\ *+<10pt>\txt{$A\otimes A$} \ar_{m}[dr] & & *+<10pt>\txt{$A\otimes A$} \ar^{m}[dl] \\ & *+<10pt>\txt{$A$} & }\end{xy}\] The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes. The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units. And going in the opposite direction, the algebra of natural transformations from an affine group scheme to its \PMlinkname{affine 1-space}{AffineSpace} is a commutative Hopf algebra, with coalgebra structure given by dualising the group structure of the affine group scheme. Further, a commutative Hopf algebra is a cogroup object in the category of commutative algebras. % Examples \begin{example}[Algebra of functions on a finite group] Let $A = C(G)$ be the algebra of complex-valued functions on a finite group $G$ and identify $C(G\times G)$ with $A \otimes A$. Then, $A$ is a Hopf algebra with comultiplication $(\Delta(f))(x,y) = f(xy)$, counit $\varepsilon(f) = f(e)$, and antipode $(S(f))(x) = f(x^{-1})$. \end{example} \begin{example}[Group algebra of a finite group] Let $A = \Cset G$ be the complex group algebra of a finite group $G$. Then, $A$ is a Hopf algebra with comultiplication $\Delta(g) = g \otimes g$, counit $\varepsilon(g) = 1$, and antipode $S(g) = g^{-1}$. \end{example} The above two examples are dual to one another. Define a bilinear form $C(G) \otimes \Cset G \to \Cset$ by $\langle f,x \rangle = f(x)$. Then, \begin{eqnarray*} \langle fg, x \rangle & = & \langle f \otimes g, \Delta(x) \rangle, \\ \langle 1, x \rangle & = & \varepsilon(x), \\ \langle \Delta(f), x \otimes y \rangle & = & \langle f, xy \rangle, \\ \varepsilon(f) & = & \langle f, e \rangle, \\ \langle S(f), x \rangle & = & \langle f, S(x) \rangle, \\ \langle f^*, x \rangle & = & \overline{\langle f, S(x)^* \rangle}. \end{eqnarray*} \begin{example}[Polynomial functions on a Lie group] Let $A = \mathrm{Poly}(G)$ be the algebra of complex-valued polynomial functions on a complex Lie group $G$ and identify $\mathrm{Poly}(G\times G)$ with $A \otimes A$. Then, $A$ is a Hopf algebra with comultiplication $(\Delta(f))(x,y) = f(xy)$, counit $\varepsilon(f) = f(e)$, and antipode $(S(f))(x) = f(x^{-1})$. \end{example} \begin{example}[Universal enveloping algebra of a Lie algebra] Let $A = \mathcal{U}(\mathfrak{g})$ be the universal enveloping algebra of a complex Lie algebra $\mathfrak{g}$. Then, $A$ is a Hopf algebra with comultiplication $\Delta(X) = X\otimes1+1\otimes X$, counit $\varepsilon(X) = 0$, and antipode $S(X) = -X$. \end{example} The above two examples are dual to one another (if $\mathfrak{g}$ is the Lie algebra of $G$). Define a bilinear form $\mathrm{Poly}(G) \otimes \mathcal{U}(\mathfrak{g}) \to \Cset$ by $\langle f,X \rangle = \deriv{}{t}\big|_{t=0} f(\exp(tX))$.