I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

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Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

Deformation quantization of Poisson manifolds, I

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Methods Of Modern Mathematical Physics

Group representations in probability and statistics

0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call them (or, rather, a slight modification of them, see ?2) bases of PBW type, since for v = 1, they specialize to bases of U+ of the type provided by the Poincare see however ? 12.)

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Canonical bases arising from quantized enveloping algebras

here l(w) is the length of w In the case where Wis a Weyl group and q is specialized to a fixed prime power, | ~ can be interpreted as the algebra of intertwining operators of the space of functions on the flag manifold of the corresponding finite Chevalley group G(Fq) (see [loc cit, Ex 24]) Therefore, the problem of decomposing this space of functions into irreducible representations of G(Fq) is equivalent to the problem of decomposing the regular representation o f ~ | (12 It is known that, in this case, | is isomorphic to the group algebra of W; however, in general, this isomorphism cannot be defined without introducing a square root of q (see [1]) It is therefore, natural to extend the ground ring of ~ as follows For any Coxeter group (W, S) we define the Hecke algebra ~ to be J{' | A, where A is the ring of Laurent polynomials with integral coefficients in the indeterminate ql/2 Our purpose is to construct representations oL,Uf endowed with a special basis They will be defined in terms of certain graphs We define a W-graph to be a set of vertices X, with a set Y of edges (an edge is a subset of X consisting of two elements) together with two additional data: for each vertex xeX , we are given a subset I x of S and, for each ordered pair of vertices y, x such that {y, x} e Y, we are given an integer p(y, x) +0 These data are subject to the requirements (10a), (10b) below Let E be

Representations of Coxeter Groups and Hecke Algebras.

Proof of the Deligne-Langlands conjecture for Hecke algebras

Structure of representations with highest weight of infinite-dimensional Lie algebras☆

In the paper [Dr3] V. Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the problem of quantization of Lie bialgebras. This paper gives a positive answer to a number of Drinfeld's questions, using the methods and ideas of [KL]. In particular, we show the existence of a quantization for Lie bialgebras. The universality and functoriality properties of this quantization will be discussed in the second paper of this series. We plan to provide positive answers to most of the remaining questions in [Dr3] in the following papers of this series.

David Kazhdan

Papers

Representations of Coxeter Groups and Hecke Algebras.

Proof of the Deligne-Langlands conjecture for Hecke algebras

Structure of representations with highest weight of infinite-dimensional Lie algebras☆

Quantization of Lie bialgebras, II

Some inverse spectral results for negatively curved 2-manifolds