HamiltonianAlgebroid 2009-05-01 02:24:39 2009-05-01 02:30:59 Definition Hamiltonian algebroids \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} \usepackage{xypic} \usepackage{pst-plot} % define commands here \newcommand*{\abs}[1]{\left\lvert #1\right\rvert} \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}} \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \subsection{Introduction} \emph{Hamiltonian algebroids} are generalizations of the Lie algebras of canonical transformations. \begin{definition} Let $X$ and $Y$ be two vector fields on a smooth manifold $M$, represented here as operators acting on functions. Their commutator, or Lie bracket, $L$, is : \begin{align*} [X,Y](f)=X(Y(f))-Y(X(f)). \end {align*} Moreover, consider the classical configuration space $Q = \bR^3$ of a classical, mechanical system, or particle whose phase space is the cotangent bundle $T^* \bR^3 \cong \bR^6$, for which the space of (classical) observables is taken to be the real vector space of smooth functions on $M$, and with T being an element of a \PMlinkname{Jordan-Lie (Poisson) algebra}{JordanBanachAndJordanLieAlgebras} whose definition is also recalled next. Thus, one defines as in classical dynamics the \emph{Poisson algebra} as a Jordan algebra in which $\circ$ is associative. We recall that one needs to consider first a \emph{specific algebra} (defined as a vector space $E$ over a ground field (typically $\bR$ or $\bC$)) equipped with a bilinear and distributive multiplication $\circ$~. Then one defines a \emph{Jordan algebra} (over $\bR$), as a a specific algebra over $\bR$ for which: $ \begin{aligned} S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 , \end{aligned},$ for all elements $S, T$ of this algebra. Then, the usual algebraic types of morphisms automorphism, isomorphism, etc.) apply to a \PMlinkname{Jordan-Lie (Poisson) algebra}{JordanBanachAndJordanLieAlgebras} defined as a real vector space $\mathfrak A_{\bR}$ together with a \emph{Jordan product} $\circ$ and \emph{Poisson bracket} $\{~,~\}$, satisfying~: \begin{itemize} \item[1.] for all $S, T \in \mathfrak A_{\bR},$ $\begin{aligned} S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\} \end{aligned}$ \item[2.] the \emph{Leibniz rule} holds $ \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}$ for all $S, T, W \in \mathfrak A_{\bR}$, along with \item[3.] the \emph{Jacobi identity}~:\\ $ \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$ \item[4.] for some $\hslash^2 \in \bR$, there is the \emph{associator identity} ~: \bigbreak $(S \circ T) \circ W - S \circ (T \circ W) = \frac{1}{4} \hslash^2 \{\{S, W \}, T \}~.$ \end{itemize} Thus, the canonical transformations of the Poisson sigma model phase space specified by the \PMlinkname{Jordan-Lie (Poisson) algebra}{JordanBanachAndJordanLieAlgebras} (also Poisson algebra), which is determined by both the Poisson bracket and the \emph{Jordan product} $\circ$, define a \emph{Hamiltonian algebroid} with the Lie brackets $L$ related to such a Poisson structure on the target space. \end{definition}