Results 1  10
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45
Richardson: Cohomology and deformations in graded Lie algebras
 Bull. Amer. Math. Soc
, 1966
"... authors gave an outline of the similarities between the deformations of complexanalytic structures on compact manifolds on one hand, and the deformations of associative algebras on the other. The first theory had been stimulated in 1957 by a paper [7] by Nijenhuis ..."
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Cited by 44 (0 self)
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authors gave an outline of the similarities between the deformations of complexanalytic structures on compact manifolds on one hand, and the deformations of associative algebras on the other. The first theory had been stimulated in 1957 by a paper [7] by Nijenhuis
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
"... ..."
Cohomology of Algebraic Theories
 J. of Algebra
, 1991
"... this paper were announced in [11], [18]. Some of them were obtained by the second author only, in particular, the main theorem of Section 2. 2. On the Mac Lane Cohomology In this section we prove Theorem A, which is concerned with the relationship between Mac Lane cohomology of rings and Ext groups ..."
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Cited by 30 (17 self)
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this paper were announced in [11], [18]. Some of them were obtained by the second author only, in particular, the main theorem of Section 2. 2. On the Mac Lane Cohomology In this section we prove Theorem A, which is concerned with the relationship between Mac Lane cohomology of rings and Ext groups in functor categories; related questions are discussed. Let us recall the denition of the Mac Lane cohomology from [13]. Consider the sets Cn with 2 n elements  ntuples ("1 ; :::; "n ), where " i = 0 or 1, for n > 0 and i 6 n, and the 0tuple ( ) for n = 0. For convenience Cn can be visualized as the set of vertices of an ncube, the product of n copies of the 1cube with vertices 0 and 1. Dene maps 0 i ; 1 i : Cn ! Cn+1 , 1 6 i 6 n + 1, by the equalities 0 i ("1 ; :::; "n) = ("1 ; :::; " i 1 ; 0; " i+1 ; :::; "n ); 1 i ("1 ; :::; "n) = ("1 ; :::; " i 1 ; 1; " i+1 ; :::; "n ): For an abelian group A and a set S, let A[S] denote the sum of S copies of the group A. Since the sets Cn are nite, the group A[Cn ] can be identied with the group of all maps t : Cn ! A: Let Q 0 n (A) be the free abelian group generated by the set A[Cn ], i. e., Q 0 n (A) = Z[A[Cn ]]: Following Mac Lane [13], dene for i = 1; 2; :::; n the homomorphisms R i ; S i ; P i : Q 0 n (A) ! Q 0 n 1 (A) by R i = Z[R i ]; S i = Z[S i ]; P i = Z[P i ]; COHOMOLOGY OF ALGEBRAIC THEORIES 257 where R i ; S i ; P i : A[Cn ] ! A[Cn 1 ] are homomorphisms dened for e 2 Cn 1 and t 2 A[Cn ] by (R i t)(e) = t(0 i e); (S i t)(e) = t(1 i e); (P i t)(e) = t(0 i e) + t(1 i e): In [13], Mac Lane denes the boundary homomorphism @ : Q 0 n (A) ! Q 0 n 1 (A) by the equality @ = n X i=1 ( 1) i (P i R i S i ): A generator t : Cn ! A of the group Q 0 n (A) is called a slab when t( ) = 0, for n...
Deformation theory of infinity algebras
 Journal of Algebra
"... Abstract. This work explores the deformation theory of algebraic structures in a very general setting. These structures include commutative, associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy associative and Lie algebras. In all these cases the a ..."
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Abstract. This work explores the deformation theory of algebraic structures in a very general setting. These structures include commutative, associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy associative and Lie algebras. In all these cases the algebra structure is determined by an element of a certain graded Lie algebra which plays the role of a differential on this algebra. We work out the deformation theory in terms of the Lie algebra of coderivations of an appropriate coalgebra structure and construct a universal infinitesimal deformation as well as a miniversal formal deformation. By working at this level of generality, the main ideas involved in deformation theory stand out more clearly. 1.
On finitedimensional semisimple and cosemisimple Hopf algebras in positive characteristic
 Internat. Math. Res. Notices
, 1998
"... Recently, important progress has been made in the study of finitedimensional semisimple Hopf algebras over a field of characteristic zero (see [Mo] and references therein). Yet, very little is known over a field k of positive characteristic. In this paper we first prove in Theorem 2.1 that any fini ..."
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Cited by 18 (6 self)
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Recently, important progress has been made in the study of finitedimensional semisimple Hopf algebras over a field of characteristic zero (see [Mo] and references therein). Yet, very little is known over a field k of positive characteristic. In this paper we first prove in Theorem 2.1 that any finitedimensional semisimple and
A∞ algebras and the cohomology of moduli spaces
 Trans. Amer. Math. Soc
, 1995
"... Let us consider an A ∞ algebra with an invariant inner product. The main goal of this paper is to classify the infinitesimal deformations of this A ∞ algebra preserving the inner product and to apply this result to the construction of homology classes on the moduli spaces of algebraic curves. With t ..."
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Cited by 18 (6 self)
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Let us consider an A ∞ algebra with an invariant inner product. The main goal of this paper is to classify the infinitesimal deformations of this A ∞ algebra preserving the inner product and to apply this result to the construction of homology classes on the moduli spaces of algebraic curves. With this aim, we define cyclic cohomology
Infinity algebras and the homology of graph complexes, preprint qalg/9601018
"... Abstract. An L ∞ algebra is a generalization of a Lie algebra [10, 9, 17]. Given an L ∞ algebra with an invariant inner product, we construct a cycle in the homology of the complex of metric ordinary graphs. Since the cyclic cohomology of a Lie algebra determines infinitesimal deformations of the al ..."
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Cited by 10 (4 self)
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Abstract. An L ∞ algebra is a generalization of a Lie algebra [10, 9, 17]. Given an L ∞ algebra with an invariant inner product, we construct a cycle in the homology of the complex of metric ordinary graphs. Since the cyclic cohomology of a Lie algebra determines infinitesimal deformations of the algebra into an L ∞ algebra, this construction shows that a cyclic cocycle of a Lie algebra determines a cycle in the homology of the graph complex. This result was suggested by a remark by M. Kontsevich in [7] that every finite dimensional Lie algebra with an invariant inner product determines a cycle in the graph complex. In a joint paper with A. Schwarz [14], we proved that an A∞ algebra with an invariant inner product determines a cycle in the homology of the complex of metric ribbon graphs. In this article, a simpler proof of this fact is given. Both constructions are based on the ideas presented in [13]. 1.
When the theories meet: Khovanov homology as Hochschild homology of links, arXiv:math.GT/0509334
"... ABSTRACT. We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we ..."
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Cited by 10 (3 self)
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ABSTRACT. We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of KhovanovRozansky, sl(n), homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplicationfree version of Khovanov homology for graphs developed by L. HelmeGuizon and Y. Rong and extended here to Mreduced case, and in the case of a polygon to noncommutative algebras. In this framework we prove that for any unital algebra A the Hochschild homology of A is isomorphic to graph homology over A of a polygon. We expect that this
MANIN PRODUCTS, KOSZUL DUALITY, LODAY ALGEBRAS AND DELIGNE CONJECTURE
"... Dedicated to JeanLouis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, nonsymmetric operads, operads, ..."
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Dedicated to JeanLouis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, nonsymmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne’s conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne’s conjecture.
INVITATION TO COMPOSITION
, 2001
"... Abstract. In 1963 [Ann. of Math. 78, 267288], Gerstenhaber invented a comp(osition) calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp system) setting. In particular, as in the Hochschild complex, ..."
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Cited by 8 (7 self)
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Abstract. In 1963 [Ann. of Math. 78, 267288], Gerstenhaber invented a comp(osition) calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp system) setting. In particular, as in the Hochschild complex, a graded Lie algebra and a precoboundary operator can be associated to every comp system. A derivation deviation of the precoboundary operator over the total composition is calculated in two ways, (the long) one of which is essentially new and can be seen as an example and elaboration of the auxiliary variables method proposed by Gerstenhaber in the early days of the comp calculus. Classification. 18D50 (MSC2000). Key words. Comp(osition), (pre)operad, Gerstenhaber theory, cup, graded Lie algebra, (pre)coboundary, derivation deviation.