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An homotopy formula for the Hochschild cohomology. (English) Zbl 0842.16006

Let \(N\) be a smooth, Hausdorff, second countable manifold, let \(N\) be the space of smooth complex valued functions on \(N\), and for each integer \(a\) let \({\mathcal M}^a\) denote the space of \((a+1)\)-linear local maps from \(N\) to itself. A multilinear map \(A : N^{a+1} \to N\) is local if for all \(u_0,\dots, u_a \in N\) the support of \(A(u_0,\dots, u_a)\) is contained in the intersection of the supports of the \(u_i\); this implies that \(A\) is locally a multidifferential operator. Let \({\mathcal M}=\bigoplus_{a \in Z} {\mathcal M}^a\). \(\mathcal M\) has the structure of a \(\mathbb{Z}\)-graded Lie algebra. It can also be regarded as the Hochschild cochains of \(N\) in \(N\) whose restrictions to the relatively compact open subsets of \(M\) are differential, and the Hochschild coboundary \(\delta : {\mathcal M} \to {\mathcal M}\) can be defined from the Lie algebra structure. \(A \in {\mathcal M}^a\) is said to be \(nc\) if \(A(u_0,\dots, u_a)=0\) if some \(u_i\) is constant (or if \(a=-1\)). The subspace \({\mathcal M}_{nc}\) of \(nc\) elements in \(\mathcal M\) is a subgraded Lie algebra stable under \(\delta\) and the inclusion induces an isomorphism on cohomology. The main result of this paper is the construction of a homotopy map \(K\) on \({\mathcal M}_{nc}\); that is such that \(K \circ \delta+\delta \circ K=\text{id} - \alpha\), where \(\alpha(A)\) is the skew-symmetrization of the terms of order 1 in each argument of \(A\). They use this result to construct a star product on the dual of a Lie algebra in such a way that the original powers and the star-powers coincide, and they further show that this property characterizes the deformation.
Reviewer: A.R.Magid (Norman)

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B70 Graded Lie (super)algebras
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
17B56 Cohomology of Lie (super)algebras
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References:

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