Massey brackets and deformations (original) (raw)
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Massey Products in Graded Lie Algebra Cohomology
2005
We discuss Massey products in a N-graded Lie algebra cohomology. One of the main examples is so-called ”positive part” L1 of the Witt algebra W. Buchstaber conjectured that H ∗ (L1) is generated with respect to non-trivial Massey products by H¹(L1). Feigin, Fuchs and Retakh represented H ∗ (L1) by trivial Massey products and the second part of the Buchstaber conjecture is still open. We consider the associated graded algebra m0 of L1 with respect to the filtration by its descending central series and prove that H ∗ (m0) is generated with respect to non-trivial Massey products by H¹(m0).
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The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra sl 2 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).
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For Alex Heller on his 65th birthday Retakh, V.S.. Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras, Journal of Pure and Applied Algebra 89 (1993) 217-229. The definition of rr-homotopically multiplicative maps of differential graded Lie algebras is given. It is shown that such maps conserve n-Lie-Massey brackets.
A ] 1 1 Ja n 20 07 Higher Derived Brackets and Deformation Theory I
2008
The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order operators. We then introduce a unifying theme for building derived brackets and show that two prevalent derived Lie bracket constructions are equivalent. Two basic methods of constructing derived strict sh-Lie brackets are also shown to be essentially the same. So far, each of these derived brackets is defined on an abelian subalgebra of a Lie algebra. We describe, as an alternative, a cohomological construction of derived sh-Lie brackets. Namely, we prove that a differential algebra with a graded homotopy commutative and associative product and an odd, square-zero operator (that commutes with the differential) gives rise to an sh-Lie structure on the cohomology via derived brackets. The method is in particular applicable to differential vertex operato...
Deformations of Lie algebras. Math.USSR-Sb. 55 (1986), 467-473
The author considers general questions of deformations of Lie algebras over a field of characteristic zero, and the related problems of computing cohomology with coefficients in adjoint representations. The construction of a versal family, and the construction of obstructions to the extension of deformations, are also considered. Bibliography: 13 titles.
Notes on formal deformations of Hom-associative and Hom-Lie algebras
The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra sl 2 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).
Higher derived brackets and deformation theory I
Arxiv preprint math/0504541, 2005
The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order operators. We then introduce a unifying theme for building derived brackets and show that two prevalent derived Lie bracket constructions are equivalent. Two basic methods of constructing derived strict sh-Lie brackets are also shown to be essentially the same. So far, each of these derived brackets is defined on an abelian subalgebra of a Lie algebra. We describe, as an alternative, a cohomological construction of derived sh-Lie brackets. Namely, we prove that a differential algebra with a graded homotopy commutative and associative product and an odd, square-zero operator (that commutes with the differential) gives rise to an sh-Lie structure on the cohomology via derived brackets. The method is in particular applicable to differential vertex operator algebras.
1985
The author considers general questions of deformations of Lie algebras over a field of characteristic zero, and the related problems of computing cohomology with coefficients in adjoint representations. The construction of a versal family, and the construction of obstructions to the extension of deformations, are also considered. Bibliography: 13 titles.
A ] 2 6 A pr 2 00 5 Higher Derived Brackets and Deformation Theory I
2008
We prove the equivalence of several different definitions of higher order differential operators and define differential operators of lower (negative) orders. We then study derived Lie and sh-Lie brackets on an abelian subalgebra of a Lie algebra as well as the cohomology of a certain type of
Notes on formal deformations of Hom-associative and Hom-Lie algebras, to appear in Forum Math
The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra sl 2 (K) and describe as formal deformations the q-deformed Witt algebra and Jackson sl 2 (K).