Deformation quantization with traces and vanishing of the wheels
The paper has been withdrawn because the result of math.QA/0002057 "Deformation quantization with traces" holds only for a constant volume form.
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arXiv subjects
Boris Shoikhet
Publications and source records attributed to Boris Shoikhet.
Gerstenhaber bracket on double Hochschild complex and deformation theory
We construct a differential and a Lie bracket on the space\linebreak $\{\Hom (A^{\otimes k}, A^{\otimes l})\},_{k,l\ge 0}$ for any associative algebra $A$. The restriction of this bracket to the space $\{\Hom (A^{\otimes k}, A)\},_{k\ge 0}$ is exactly the Gerstenhaber bracket. We discuss some formality conjecture related with this construction. We also discuss some applications to deformation theory.
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On the cyclic Formality conjecture
We conjecture an explicit formula for a cyclic analog of the Formality $L_{\infty}$-morphism [K]. We prove that its first Taylor component, the cyclic Hochschild-Kostant-Rosenberg map, is in fact a morphism (and a quasiisomorphism) of the complexes. To prove it we construct a cohomological version of the Connes-Tsygan bicomplex in cyclic homology. As an application of the cyclic Formality conjecture, we obtain an explicit formula for cyclically invariant deformation quantization.We show that (a more precise version of) the Connes-Flato-Sternheimer conjecture [CFS] on the existence of closed star-products on a symplectic manifold also follows from our conjecture.
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On the Kontsevich and the Campbell-Baker-Hausdorff deformation quantizations of a linear Poisson structure
For the Kirillov-Poisson structure on the vector space $\g^*$, where $\g$ is a finite-dimensional Lie algebra, it is known at least two canonical deformations quantization of this structure: they are the M. Kontsevich universal formula [K], and the formula, arising from the classical Campbell-Baker-Hausdorff formula [Ka]. It was proved in [Ka] that the last formula is exactly the part of Kontsevich's formula consisting of all the admissible graphs without (oriented) cycles between the vertices of the first type. It follows from the CBH-theorem that this part of Kontsevich's formula defines an associative product (in the case of a linear Poisson structure). The aim of these notes is to prove the last result directly, using the methods analogous to [K] instead of the CBH-formula. We construct an $L_\infty$-morphism $\U_\lin\colon[T^\ndot_\poly]_\lin\to D^\ndot_\poly$ from the \dg Lie algebra of polyvector fields with linear coefficients to the \dg Lie algebra of polydifferential operators, which is not equal to the restriction of the Formality $L_\infty$-morphism $\U\colon T^\ndot_\poly\to D^\ndot_\poly$ [K] to the subalgebra $[T^\ndot_\poly]_\lin$. For a bivector field $α$ with linear coefficients such that $[α,α]=0$ the corresponding solution $\U_\lin(α)$ of the Maurer-Cartan equation in $D^\ndot_\poly$ defines exactly the CBH-quantization,in the case of the harmonic angle map [K], Sect.2.We prove the associativity of the restricted Kontsevich formula (in the linear case) also for any angle map [K], Sect.6.2.
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Integration of the Lifting formulas and the cyclic homology of the algebras of differential operators
We integrate the Lifting cocycles $Ψ_{2n+1},Ψ_{2n+3},Ψ_{2n+5},...$ ([Sh1], [Sh2]) on the Lie algebra $\Dif_n$ of holomorphic differential operators on an $n$-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle $λ$ on an $n$-dimensional complex manifold $M$ in the sense of Gelfand-Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin-Tsygan theorem [FT1]: $H^\bullet_\Lie(\gl^\fin_\infty(\Dif_n);\C) = \wedge^\bullet(Ψ_{2n+1}, Ψ_{2n+3},Ψ_{2n+5}, ...)$.
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On the Duflo formula for $L_\infty$-algebras and Q-manifolds
We prove a direct analogue of the classical Duflo formula in the case of $L_\infty$-algebras. We conjecture an analogous formula in the case of an arbitrary Q-manifold. When $G$ is a compact connected Lie group, the Duflo theorem for the Q-manifold $(ΠTG,d_{DR})$ is exactly the Duflo theorem for the Lie algebra $g = Lie G$. The corresponding theorem for the Q-manifold $(ΠTM,d_{DR})$, where $M$ is an arbitrary smooth manifold, is a generalization of the Duflo theorem for the case of smooth manifolds. On the other hand, the Duflo theorem for the Q-manifold $(Π\bar T_{hol} M, \bar\partial)$, where $M$ is a complex manifold, is a generalization of the M. Kontsevich's ``theorem on complex manifold'' [K1], Sect. 8.4.
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On the $A_\infty$-Formality conjecture
It is proved that the associative differential graded algebra of (polynomial) polyvector fields on a vector space (may be infinite- dimensional) is quasi-isomorphic to the corresponding cohomological Hochschild complex of (polynomial) functions on this vector space as an associative differential graded algebra. This result is an $A_\infty$-version of the Formality conjecture of Maxim Kontsevich [K].
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Lifting formulas II
In the present paper we generalize the lifting formulas from [Sh] and obtain the formula for the (2n+2l-1)-cocycle on the LIe algebra of differential operators on a n-dimensional space for arbitrary n and l.
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Cohomology of the Lie algebras of differential operators: lifting formulas
We construct the explicit formula for the (2n+1)-cocycle of the Lie algebra of (pseudo)differential operators on a n-dimensional space. We prove that this formula in fact defines a cocycle for n=1 and n=2.
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Representations of the Lie algebra gl(infinity)^ and ``reciprocity formula'' for Clebsch-Gordan coefficients
We compare two approaches to the calculation of irreducible characters of the Lie algebra gl(infinity)^ with negative integral central charge. As a consequence, we obtain a "reciprocity formula" for Clebsch-Gordan coefficients, in the limit when the central charge tends to infinity.
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