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To any Poisson manifold $(M,\pi)$ (P,\pi)$ is associated an anchor map $\pi_\sharp:T^*M\rightarrow TM$ \pi_\sharp:T^*P\longrightarrow TP$ given by $\beta(\pi_\sharp(\alpha))=\pi(\alpha,\beta)$, \beta(\pi_\sharp(\alpha))=\pi(\alpha,\beta),$ and a Lie bracket on $1$-forms $[alpha,\beta]=\mathscr{L}{\pi\sharp(\alpha)}\beta-\mathscr{L}{\pi$ [\alpha,\beta]= \sharp(\beta)}\alpha-d(\pi(\alpha,\beta)).$mathscr{L}_{\pi_{\#}(\alpha)}\beta-\mathscr{L}_{\pi_{\#}(\beta)}\alpha-d(\pi(\alpha,\beta)). $$ A contravariant connection on $(M,\pi)$ (P,\pi)$ is an $\mathbb{R}$-bilinear map $\begin{array}{cccc} $ \begin{array}{cccc} \mathcal{D} : &\Omega^1(M)\times\Omega^1(M)&\longrightarrow&\Omega^1(M) &\Omega^1(P)\times\Omega^1(P)&\longrightarrow&\Omega^1(P) \cr &(\alpha,\beta)&\longmapsto&\mathcal{D}_\alpha\beta \end{array}$end{array} $$ such that, for any $f\in C^\infty(M)$

$\mathcal{D}{f\alpha}\beta=f\mathcal{D}\alpha\beta$, \mathcal{D}_{f\alpha}\beta=f\mathcal{D}_\alpha\beta$,

$\mathcal{D}\alpha\left(f\beta\right)=f\mathcal{D}\alpha\beta+\pi_\sharp(\alpha)(f)\beta$.\mathcal{D}_\alpha\left(f\beta\right)=f\mathcal{D}_\alpha\beta+\pi_\sharp(\alpha)(f)\beta$.

The torsion and the curvature of a contravariant connection $\mathcal{D}$ {\cal D}$ are formally identical to the usual definitions

$T(\alpha,\beta)={\cal D}\alpha\beta-{\cal D}\beta\alpha-[\alpha,\beta]$ and $$T(\alpha,\beta)={\cal D}_\alpha\beta-{\cal D}_\beta\alpha-[\alpha,\beta]\quad\mbox{and}\quad K(\alpha,\beta)={\cal D}\alpha{\cal D}\beta-{\cal D}\beta{\cal D}\alpha-{\cal D}_{[\alpha,\beta]}$.D}_\alpha{\cal D}_\beta-{\cal D}_\beta{\cal D}_\alpha-{\cal D}_{[\alpha,\beta]}.$$ Eli Hawkins showed in http://arxiv.org/abs/math/0504232here that if $0 $0 \to \hbar\Omega^* \to \Omega^* \stackrel{\cal P}\to \Omega^*(M) \to 0$0$$ is a deformation of $\Omega^(M)$, \Omega^*(M)$, the differential graded algebra of differential forms, then we can define a generalized Poisson bracket on $\Omega^(M)$ \Omega^*(M)$ by $\lbrace\alpha,\beta\rbrace $ \{\alpha,\beta\} = \mathcal{P}\left(\frac{i}{\hbar} [\hat\alpha,\hat\beta]\right)$.\hat\alpha,\hat\beta]\right).$$ This Poisson bracket defines a contravariant connection: $\lbrace f,\alpha\rbrace=\mathcal{D}_{df}\alpha$ {f,\alpha}=\mathcal{D}_{df}\alpha$ and a metacurvature tensor , a symetric $(2,3)$-tensor $\mathcal{M}$, given by $\mathcal{M}(df,\alpha,\beta)=\lbrace f,\lbrace\alpha,\beta\rbrace\rbrace-\lbrace\lbrace f,\alpha\rbrace,\beta\rbrace-\lbrace\lbrace f,\beta\rbrace,\alpha\rbrace$.$\mathcal{M}(df,\alpha,\beta)=\{f,\{\alpha,\beta\}\}-\{\{f,\alpha\},\beta\}- \{\{f,\beta\},\alpha\}.$$ The generalized Poisson bracket staisfies the graded Jacobi identity if, and only if, the curvature and the metacurvature vanish metcurvature vanishe identically. My problem is to study the metacurvature tensor, for which there is no formula except for the symplectic case (Theorem 2.4). I was suggested to use technics from super calculus. I do not know yet what it is, may you suggest me some nice references? Thank you!
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To any Poisson manifold $(P,\pi)$ (M,\pi)$ is associated an anchor map $\pi_\sharp:T^*P\longrightarrow TP$ \pi_\sharp:T^*M\rightarrow TM$ given by $\beta(\pi_\sharp(\alpha))=\pi(\alpha,\beta),$ \beta(\pi_\sharp(\alpha))=\pi(\alpha,\beta)$, and a Lie bracket on $1$-forms

$$ [\alpha,\beta]= [alpha,\beta]=\mathscr{L}{\pi\mathscr{L}_{\pi_{\#}(\alpha)}\beta-\mathscr{L}_{\pi_{\#}(\beta)}\alpha-d(\pi(\alpha,\beta)). $$sharp(\alpha)}\beta-\mathscr{L}{\pi\sharp(\beta)}\alpha-d(\pi(\alpha,\beta)).$

A contravariant connection on $(P,\pi)$ (M,\pi)$ is an $\mathbb{R}$-bilinear map

$$ \begin{array}{cccc} \begin{array}{cccc} \mathcal{D} : &\Omega^1(P)\times\Omega^1(P)&\longrightarrow&\Omega^1(P) &\Omega^1(M)\times\Omega^1(M)&\longrightarrow&\Omega^1(M) \ cr &(\alpha,\beta)&\longmapsto&\mathcal{D}_\alpha\beta \end{array} $$end{array}$

such that, for any $f\in C^\infty(M)$

$\mathcal{D}_{f\alpha}\beta=f\mathcal{D}_\alpha\beta$,\mathcal{D}{f\alpha}\beta=f\mathcal{D}\alpha\beta$, $\mathcal{D}_\alpha\left(f\beta\right)=f\mathcal{D}_\alpha\beta+\pi_\sharp(\alpha)(f)\beta$.\mathcal{D}\alpha\left(f\beta\right)=f\mathcal{D}\alpha\beta+\pi_\sharp(\alpha)(f)\beta$.

The torsion and the curvature of a contravariant connection ${\cal D}$ \mathcal{D}$ are formally identical to the usual definitions

$$T(\alpha,\beta)={\cal D}_\alpha\beta-{\cal D}_\beta\alpha-[\alpha,\beta]\quad\mbox{and}\quad T(\alpha,\beta)={\cal D}\alpha\beta-{\cal D}\beta\alpha-[\alpha,\beta]$ and $K(\alpha,\beta)={\cal D}_\alpha{\cal D}_\beta-{\cal D}_\beta{\cal D}_\alpha-{\cal D}_{[\alpha,\beta]}.$$D}\alpha{\cal D}\beta-{\cal D}\beta{\cal D}\alpha-{\cal D}_{[\alpha,\beta]}$.

Eli Hawkins showed herein http://arxiv.org/abs/math/0504232 that if

$$0 0 \to \hbar\Omega^* \to \Omega^* \stackrel{\cal P}\to \Omega^*(M) \to 0$$0$

is a deformation of $\Omega^*(M)$, \Omega^(M)$, the differential graded algebra of differential forms, then we can define a generalized Poisson bracket on $\Omega^*(M)$\Omega^(M)$ by$

$\{\alpha,\beta\} \lbrace\alpha,\beta\rbrace = \mathcal{P}\left(\frac{i}{\hbar} [\hat\alpha,\hat\beta]\right).$$\hat\alpha,\hat\beta]\right)$.

This Poisson bracket defines a contravariant connection: ${f,\alpha}=\mathcal{D}_{df}\alpha$ \lbrace f,\alpha\rbrace=\mathcal{D}_{df}\alpha$ and a metacurvature tensor, a symetric $(2,3)$-tensor $\mathcal{M}$, given by

$$\mathcal{M}(df,\alpha,\beta)=\{f,\{\alpha,\beta\}\}-\{\{f,\alpha\},\beta\}- \{\{f,\beta\},\alpha\}.$$\mathcal{M}(df,\alpha,\beta)=\lbrace f,\lbrace\alpha,\beta\rbrace\rbrace-\lbrace\lbrace f,\alpha\rbrace,\beta\rbrace-\lbrace\lbrace f,\beta\rbrace,\alpha\rbrace$.

The generalized Poisson bracket staisfies the graded Jacobi identity if, and only if, the curvature and the metcurvature vanishe metacurvature vanish identically. My problem is to study the metacurvature tensor, for which there is no formula except for the symplectic case (Theorem 2.4). I was suggested to use technics from super calculus. I do not know yet what it is, may you suggest me some nice references? Thank you!
show/hide this revision's text 2 fixed LaTeX; added 2 characters in body

To any Poisson manifold $(P,\pi)$ is associated an anchor map $\pi_\sharp:T^*P\longrightarrow TP$ given by $\beta(\pi_\sharp(\alpha))=\pi(\alpha,\beta),$ and a Lie bracket on $1$-forms \begin{equation} $$ [\alpha,\beta]= \mathscr{L}{\pi{#}(\alpha)}\beta-\mathscr{L}{\pi{#}(\beta)}\alpha-d(\pi(\alpha,\beta)). \end{equation} mathscr{L}_{\pi_{\#}(\alpha)}\beta-\mathscr{L}_{\pi_{\#}(\beta)}\alpha-d(\pi(\alpha,\beta)). $$ A contravariant connection on $(P,\pi)$ is an $\mathbb{R}$-bilinear map \begin{equation*} $$ \begin{array}{cccc} \mathcal{D} : &\Omega^1(P)\times\Omega^1(P)&\longrightarrow&\Omega^1(P) \cr &(\alpha,\beta)&\longmapsto&\mathcal{D}_\alpha\beta \end{array} \end{equation*} $$ such that, for any $f\in C^\infty(M)$\begin{enumerate} \item

$\mathcal{D}{f\alpha}\beta=f\mathcal{D}\alpha\beta$, \item \mathcal{D}_{f\alpha}\beta=f\mathcal{D}_\alpha\beta$,

$\mathcal{D}\alpha\left(f\beta\right)=f\mathcal{D}\alpha\beta+\pi_\sharp(\alpha)(f)\beta$. \end{enumerate} \mathcal{D}_\alpha\left(f\beta\right)=f\mathcal{D}_\alpha\beta+\pi_\sharp(\alpha)(f)\beta$.

The torsion and the curvature of a contravariant connection ${\cal D}$ are formally identical to the usual definitions $$T(\alpha,\beta)={\cal D}\alpha\beta-{\cal D}\beta\alpha-[\alpha,\beta]\quad\mbox{and}\quad D}_\alpha\beta-{\cal D}_\beta\alpha-[\alpha,\beta]\quad\mbox{and}\quad K(\alpha,\beta)={\cal D}\alpha{\cal D}\beta-{\cal D}\beta{\cal D}\alpha-{\cal D}_\alpha{\cal D}_\beta-{\cal D}_\beta{\cal D}_\alpha-{\cal D}_{[\alpha,\beta]}.$$ Eli Hawkins showed in \url{http://arxiv.org/abs/math/0504232} here that if $$0 \to \hbar\Omega^* \to \Omega^* \stackrel{\cal P}\to \Omega^(M) Omega^*(M) \to 0$$ is a deformation of $\Omega^(M)$, \Omega^*(M)$, the differential graded algebra of differential forms, then we can define a generalized Poisson bracket on $\Omega^*(M)$ by [ {\alpha,\beta} $$ \{\alpha,\beta\} = \mathcal{P}\left(\frac{i}{\hbar} [\hat\alpha,\hat\beta]\right).] \hat\alpha,\hat\beta]\right).$$ This Poisson bracket defines a contravariant connection: ${f,\alpha}=\mathcal{D}_{df}\alpha$ and a \textbf{metacurvature metacurvature tensor } a symetric $(2,3)$-tensor $\mathcal{M}$, given by $$\mathcal{M}(df,\alpha,\beta)=\{f,\{\alpha,\beta\}\}-\{\{f,\alpha\},\beta\}- \begin{equation}\mathcal{M}(df,\alpha,\beta)={f,{\alpha,\beta}}-{{f,\alpha},\beta}- {{f,\beta},\alpha}.\end{equation} {\{f,\beta\},\alpha\}.$$ The generalized Poisson bracket staisfies the graded Jacobi identity if, and only if, the curvature and the metcurvature vanishe identically. My problem is to study the metacurvature tensor, for which there is no formula except for the symplectic case (Theorem 2.4). I was suggested to use technics from \emph{super calculus}super calculus. I do not know yet what it is, may you suggest me some nice references? Thank you!
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