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Operads and motives in deformation quantization
 LETTERS IN MATH.PHYS
, 1999
"... The algebraic world of associative algebras has many deep connections with the geometric world of twodimensional surfaces. Recently D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Delig ..."
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Cited by 177 (1 self)
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The algebraic world of associative algebras has many deep connections with the geometric world of twodimensional surfaces. Recently D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Deligne’s conjecture on Hochschild complexes follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois group acts on deformation quantizations, and speculate on possible relations of higherdimensional algebras and of motives to quantum field theories.
Solutions of the quantum dynamical YangBaxter equation and dynamical quantum groups
"... Abstract. The quantum dynamical YangBaxter (QDYB) equation is a useful generalization of the quantum YangBaxter (QYB) equation. This generalization was introduced by Gervais, Neveu, and Felder. Unlike the QYB equation, the QDYB equation is not an algebraic but a difference equation, with respect t ..."
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Cited by 64 (5 self)
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Abstract. The quantum dynamical YangBaxter (QDYB) equation is a useful generalization of the quantum YangBaxter (QYB) equation. This generalization was introduced by Gervais, Neveu, and Felder. Unlike the QYB equation, the QDYB equation is not an algebraic but a difference equation, with respect to a matrix function rather than a matrix. The QDYB equation and its quasiclassical analogue (the classical dynamical YangBaxter equation) arise in several areas of mathematics and mathematical physics (conformal field theory, integrable systems, representation theory). The most interesting solution of the QDYB equation is the elliptic solution, discovered by Felder. In this paper, we prove the first classification results for solutions of the QDYB equation. These results are parallel to the classification of solutions of the classical dynamical YangBaxter equation, obtained in our previous paper. All solutions we found can be obtained from Felder’s elliptic solution by a limiting process and gauge transformations. Fifteen years ago the quantum YangBaxter equation gave rise to the theory of quantum groups. Namely, it turned out that the language of quantum groups (Hopf algebras) is the adequate algebraic language to talk about solutions of the quantum YangBaxter equation. In this paper we propose a similar language, originating from Felder’s ideas, which we found to be adequate for the dynamical YangBaxter equation. This is the language of dynamical quantum groups (or hHopf algebroids), which is the quantum counterpart of the language of dynamical Poisson groupoids, introduced in our previous paper.
Noncommutative differential calculus, homotopy . . .
, 2000
"... We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures. ..."
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Cited by 57 (1 self)
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We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures.
Explicit quantization of dynamical rmatrices for finite dimensional semisimple . . .
, 1999
"... ..."
Tamarkin’s proof of Kontsevich formality theorem
 Forum Math
"... 1.1. This is an extended version of lectures given at Luminy colloquium “Quantification par déformation ” held at December, 1999. In this note we explain Tamarkin’s proof [T] of the following affine algebraic version of Kontsevich’s formality theorem. ..."
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Cited by 37 (3 self)
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1.1. This is an extended version of lectures given at Luminy colloquium “Quantification par déformation ” held at December, 1999. In this note we explain Tamarkin’s proof [T] of the following affine algebraic version of Kontsevich’s formality theorem.
Vertex algebras and vertex Poisson algebras
 Commun. Contemp. Math
, 2004
"... This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex poisson algebra are revisited and certain general construction theorems of vertex poisson algebras are given. A ..."
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Cited by 30 (1 self)
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This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex poisson algebra are revisited and certain general construction theorems of vertex poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it is proved that the associated graded vector space of a filtered vertex algebra is naturally a vertex poisson algebra. For any vertex algebra V, a general construction and a classification of good filtrations are given. To each Ngraded vertex algebra V = ∐ n∈N V (n) with V (0) = C1, a canonical (good) filtration is associated. Furthermore, a notion of ∗deformation of a vertex (poisson) algebra is formulated and a ∗deformation of vertex poisson algebras associated with vertex Lie algebras is constructed. 1
Deformation quantization of Kähler manifolds
 L. D. FADDEEV’S SEMINAR ON MATHEMATICAL PHYSICS
, 2000
"... We present an explicit formula for the deformation quantization on Kähler manifolds. ..."
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Cited by 24 (0 self)
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We present an explicit formula for the deformation quantization on Kähler manifolds.