Formal Deformations of the Poisson Lie Algebra of a Symplectic Manifold and Star-Products. Existence, Equivalence, Derivations

In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the * -representation theory of such * -algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C * -algebras. Throughout this paper the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C * -algebras, we show that the GNS construction of * -representations can be understood as Rieffel induction and, moreover, that formal Morita equivalence of two * -algebras, which is defined by the existence of a bimodule with certain additional structures, implies the equivalence of the categories of strongly non-degenerate * -representations of the two * -algebras. We discuss various examples like finite rank operators on pre-Hilbert spaces and matrix algebras over * -algebras. Formal Morita equivalence is shown to imply Morita equivalence in the ring-theoretic framework. Finally we apply our considerations to deformation theory and in particular to deformation quantization and discuss the classical limit and the deformation of equivalence bimodules. *

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“…being C-linear maps, and an analogous statement holds for multilinear maps as well, see e.g [33,. Prop.…”

mentioning

confidence: 79%

Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization

Bursztyn¹,

Waldmann²

2001

Journal of Geometry and Physics

105

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“…Recall that an associative operation * : S ⊗ S → S[[i ]] is called a starproduct [1,7,10] if it is of the form…”

Section: Quantization In the Case δ =mentioning

confidence: 99%

Conformally equivariant quantization: existence and uniqueness

Duval¹,

Lecomte²,

Ovsienko³

1999

Annales De L’institut Fourier

181

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We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold (M, g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T * M and of differential operators on tensor densities over M , both viewed as modules over the Lie algebra o(p + 1, q + 1) where p + q = dim(M ). This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of half-densities, we obtain a conformally invariant star-product.

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“…The latter was originally defined in [16] and has been used in the deformation theory of algebras [13], [17], [29]. For two cochains f 1 ∈ C k1 and f 2 ∈ C k2 one defines…”

Section: Corollary If M Is Finite or X Has The Extension Property Tmentioning

confidence: 99%

On the homology of algebras of Whitney functions over subanalytic sets

Brasselet

Pflaum

2008

Ann. Math.

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In this article we study several homology theories of the algebra E ∞ (X) of Whitney functions over a subanalytic set X ⊂ R n with a view towards noncommutative geometry. Using a localization method going back to Teleman we prove a Hochschild-Kostant-Rosenberg type theorem for E ∞ (X), when X is a regular subset of R n having regularly situated diagonals. This includes the case of subanalytic X. We also compute the Hochschild cohomology of E ∞ (X) for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories. It is shown that the periodic cyclic homology coincides with the de Rham cohomology, thus generalizing a result of Feigin-Tsygan. Motivated by the algebraic de Rham theory of Grothendieck we finally prove that for subanalytic sets the de Rham cohomology of E ∞ (X) coincides with the singular cohomology. For the proof of this result we introduce the notion of a bimeromorphic subanalytic triangulation and show that every bounded subanalytic set admits such a triangulation.

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Formal Deformations of the Poisson Lie Algebra of a Symplectic Manifold and Star-Products. Existence, Equivalence, Derivations

Cited by 32 publications

References 9 publications

Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization

Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization

Conformally equivariant quantization: existence and uniqueness

On the homology of algebras of Whitney functions over subanalytic sets

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