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DG coalgebras as formal stacks
 J. Pure Appl. Algebra
"... 1.1. In this paper we provide the category of unital (dg, unbounded) coalgebras dgcu(k) ..."
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1.1. In this paper we provide the category of unital (dg, unbounded) coalgebras dgcu(k)
Lie theory for nilpotent L∞algebras
 Ann. Math
"... Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric n ..."
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Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric nsimplex ∆ n. In [20], Sullivan reformulated Quillen’s
Deformation Quantization in Algebraic Geometry
, 2003
"... We study deformation quantization of Poisson algebraic varieties. Using the universal deformation formulas of Kontsevich, and an algebrogeometric approach to the bundle of formal coordinate systems over a smooth variety X, we prove existence of deformation quantization of the sheaf OX (assuming t ..."
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We study deformation quantization of Poisson algebraic varieties. Using the universal deformation formulas of Kontsevich, and an algebrogeometric approach to the bundle of formal coordinate systems over a smooth variety X, we prove existence of deformation quantization of the sheaf OX (assuming the vanishing of certain cohomologies). Under slightly stronger assumptions we can classify all such deformations.
ON GLOBAL DEFORMATION QUANTIZATION IN THE ALGEBRAIC CASE
, 2006
"... We give a proof of Yekutieli’s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology. ..."
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We give a proof of Yekutieli’s global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology.
Deformations of sheaves of algebras
 Adv. Math
, 2005
"... Abstract. A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformations of a scheme and equivariant defo ..."
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Abstract. A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformations of a scheme and equivariant deformations are considered. The construction is based upon a model structure on the category of presheaves which should be of an independent interest. 0.1. In this paper we study formal deformations of sheaves of algebras. The most obvious (and very important) example is that of deformations of a scheme X over a field k of characteristic zero. In two different cases, the first when X is smooth, and the second when X is affine, the description is wellknown. In both cases there is a differential graded
Deformations of homotopy algebras
 Comm. Alg
"... 1.1. Let k be a field of characteristic zero, O be a dg operad over k and let A be an Oalgebra. In this note we define formal deformations of A, construct the deformation functor DefA: dgart ≤0 (k) → ∆ 0 Ens ..."
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1.1. Let k be a field of characteristic zero, O be a dg operad over k and let A be an Oalgebra. In this note we define formal deformations of A, construct the deformation functor DefA: dgart ≤0 (k) → ∆ 0 Ens
Cohomology theories for homotopy algebras and noncommutative geometry
, 2007
"... This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞, C∞ and L∞algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodg ..."
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Cited by 5 (3 self)
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This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A∞, C∞ and L∞algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of C∞algebras. This generalizes and puts in a conceptual framework previous work by Loday and GerstenhaberSchack.
KATZ’S MIDDLE CONVOLUTION ALGORITHM
, 2006
"... Abstract. This is an expository account of Katz’s middle convolution operation on local systems over P 1 − {q1,...,qn}. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, DettweilerReiter, Harao ..."
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Abstract. This is an expository account of Katz’s middle convolution operation on local systems over P 1 − {q1,...,qn}. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, DettweilerReiter, HaraokaYokoyama. Kostov’s program for applying the Katz algorithm is to say that in the range where middle convolution no longer reduces the rank, one should give a direct construction of local systems. This has been done by Kostov and CrawleyBoevey. We describe here an alternative construction using the notion of cyclotomic harmonic bundles: these are like variations of Hodge structure except that the Hodge decomposition can go around in a circle. 1.
D.: The L∞deformation complex of diagrams of algebras
 Cambridge Studies in Advanced Mathematics
, 1994
"... Abstract. The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L∞algebra which induces a graded Lie bracket on cohomology. As an example, the L∞algebra structure on t ..."
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Abstract. The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L∞algebra which induces a graded Lie bracket on cohomology. As an example, the L∞algebra structure on the deformation complex of an associative algebra morphism g is constructed. Another example is the deformation complex of a Lie algebra morphism. The last example is the diagram describing two mutually inverse morphisms of vector spaces. Its L∞deformation complex has nontrivial l0term. Explicit formulas for the L∞operations in the above examples are given. A typical deformation complex of a diagram of algebras is a fullyfledged L∞algebra with nontrivial higher operations. Contents