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Fedosov dg manifolds associated with Lie pairs. (English) Zbl 1458.58005

This paper is on constructing a graded differential manifold associated with a Lie pair. This manifold, which is called Fedosov dg manifold, can be formed in two ways.
\(\bullet\) First construction: Let \((L,A)\) be a Lie pair, i. e., \(L\) is a Lie algebroid over a field \(\mathbb{K}\) and \(A\hookrightarrow L\) is a sub Lie algebroid of \(L\). Assume also \(B=L/A\).
Proposition [M. Stiénon and P. Xu, this paper]. Given a Lie pair \((L,A)\) with quotient \(B=L/A\), the choice of (1) a splitting \(j:L/A\rightarrow L\) of the short exact sequence \(0\rightarrow A \rightarrow L \rightarrow B \rightarrow 0\) and (2) an \(L\)-connection \(\triangledown\) on \(B\) extending the Both representation determines an operator \[d_L^{\tilde{\triangledown}}:\Gamma(\wedge^\bullet L^{\lor}\otimes \hat{S}(B^{\lor}))\rightarrow \Gamma(\wedge^{\bullet+1} L^{\lor}\otimes \hat{S}(B^{\lor})),\] making \((L[1]\oplus B, d_L^{\tilde{\triangledown}})\) a dg manifold, where \(X^\lor\) denotes the dual space of \(X\).
Remark. The dg manifolds \((L[1]\oplus B, d_L^{\tilde{\triangledown}})\) do not depend on the choice of the connection and the splitting. Means, if \(j_1,j_2\) be two splittings \(B\rightarrow L\) and \(\triangledown_1, \triangledown_2\) be torsion free \(L\)-connections on \(B\). Then one can show that \((L[1]\oplus B, d_L^{\tilde{\triangledown_1}})\) and \((L[1]\oplus B, d_L^{\tilde{\triangledown_2}})\) are isomorphic.
\(\bullet\) Second construction: Consider the bundle of the graded commutative algebra \(\wedge^{\bullet}B^{\lor}\otimes \hat{S}(B^{\lor})\) and let \(\dot{\delta}\) be a fiberwise derivation of degree \((+1)\) which acts on the generators of \(\wedge^{\bullet}B^{\lor}\otimes \hat{S}(B^{\lor})\) as follows \[\dot{\delta}(1\otimes\chi)=\chi\otimes 1, \quad \dot{\delta}(\chi\otimes 1)=0, \quad \forall \chi\in B^{\lor}.\] Let also \(\dot{D}\) be a fiberwise connection of degree \((-1)\) with the following action on the generators of \(\wedge^{\bullet}B^{\lor}\otimes \hat{S}(B^{\lor})\), \[ \dot{D}(1\otimes\chi)=0, \quad \dot{D}(\chi\otimes 1)=\chi\otimes 1, \quad \forall \chi\in B^{\lor}. \] Proposition [M. Stiénon and P. Xu, this paper]. Let \((L,A)\) be a Lie pair. Consider a splitting \(i\circ p+j\circ q = id_L\) of the short exact sequence \[0\stackrel{i}{\rightarrow}L \stackrel{q}{\rightarrow} B \rightarrow 0,\] where \(p:L\rightarrow A\) and \(q:B\rightarrow L\). Let \(\triangledown\) be a torsion free \(L\)-connection on \(B\). Then there exists a unique \(1\)-form valued in the formal vertical vector bundles on \(B\): \[X^{\triangledown}\in \Gamma(\wedge^1 L^{\lor}\otimes \hat{S}^{\geq 2}(B^{\lor})\otimes B),\] defined by \[Q=-\delta+d_L^{\triangledown}+X^{\triangledown}\] that satisfies \(Q^2=0\). Here \(X^{\triangledown}\) acts on the algebra \(\Gamma(\wedge^\bullet L^\lor\otimes\hat{S}(B^\lor))\) as a derivation in a natural fashion. As a consequence \((L[1]\oplus B, Q)\) is a dg manifold.
The following theorem extends the Dolgushev-Fedosov quasi isomorphism to the context of Lie pairs. Theorem [M. Stiénon and P. Xu, this paper]. Given a Lie pair \((L,A)\), let \(d_L^{\tilde{\triangledown}}\) be the homological vector field on \(L[1]\oplus B\) determined by the choice of a splitting \(i\circ p+j\circ q=id_L\) of the short exact sequence \[0\stackrel{i}{\rightarrow}L \stackrel{q}{\rightarrow} B \rightarrow 0,\] and an \(L\)-connection \(\triangledown\) on \(B\) as above. Then the natural inclusion \(A[1],d_A\hookrightarrow (L[1]\oplus B, d_L^{\tilde{\triangledown}})\) is a quasi-isomorphism of dg manifolds.
As an application, in this paper, a simple proof of the following theorem, which is the Theorem 10 in [C. Emmrich and A. Weinstein, Prog. Math. 123, 217–239 (1994; Zbl 0846.58031)], is given.
Theorem [C. Emmrich and A. Weinstein]. Given a torsion free affine connection \(\triangledown\) on \(M\), the ‘formal exponential map’ \(EXP:(T_M)_\infty\rightarrow M\) coincides with the infinite-order jet of the geodesic exponential map \(exp\) determined by the connection \(\triangledown\).

MSC:

58A50 Supermanifolds and graded manifolds
58H05 Pseudogroups and differentiable groupoids
17B70 Graded Lie (super)algebras

Citations:

Zbl 0846.58031
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References:

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