dorsal/arxiv
View SchemaL-infinity algebras and their cohomology
| Authors | Michael Penkava |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9512014 |
| URL | https://arxiv.org/abs/q-alg/9512014 |
Abstract
An associative algebra is nothing but an odd quadratic codifferential on the tensor coalgebra of a vector space, and an A-infinity algebra is simply an arbitrary odd codifferential. Hochschild cohomology classifies the deformations of an associative algebra into an A-infinity algebra, and cyclic cohomology in the presence of an invariant inner product classifies the deformations of the associative algebra into an A-infinity algebra preserving the inner product. Similarly, a graded Lie algebra is simply a special case of an odd codifferential on the exterior coalgebra of a vector space, and an L-infinity algebra is a more general codifferential. In this paper, ordinary and cyclic cohomology of L-infinity algebras is defined, and it is shown that the cohomology of a Lie algebra (with coefficients in the adjoint representation) classifies the deformations of the Lie algebra into an L-infinity algebra. Similarly, the cyclic cohomology of a Lie algebra with an invariant inner product classifies the deformations of the Lie algebra into an L-infinity algebra which preserve the invariant inner product. The exterior coalgebra of a vector space is dual to the symmetric coalgebra of the parity reversion of the space, while the tensor coalgebra of a vector space is dual to the tensor coalgebra of its parity reversion. Using this duality, we introduce a modified bracket in the space of coderivations of the tensor and exterior coalgebras which makes it possible to treat the cohomology of an A-infinity or L-infinity as a differential graded algebra in the same manner in which the Gerstenhaber bracket is used to transform the Hochschild cochains of an associative algebra into a differential graded algebra.
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"abstract": "An associative algebra is nothing but an odd quadratic codifferential on the\ntensor coalgebra of a vector space, and an A-infinity algebra is simply an\narbitrary odd codifferential. Hochschild cohomology classifies the deformations\nof an associative algebra into an A-infinity algebra, and cyclic cohomology in\nthe presence of an invariant inner product classifies the deformations of the\nassociative algebra into an A-infinity algebra preserving the inner product.\nSimilarly, a graded Lie algebra is simply a special case of an odd\ncodifferential on the exterior coalgebra of a vector space, and an L-infinity\nalgebra is a more general codifferential. In this paper, ordinary and cyclic\ncohomology of L-infinity algebras is defined, and it is shown that the\ncohomology of a Lie algebra (with coefficients in the adjoint representation)\nclassifies the deformations of the Lie algebra into an L-infinity algebra.\nSimilarly, the cyclic cohomology of a Lie algebra with an invariant inner\nproduct classifies the deformations of the Lie algebra into an L-infinity\nalgebra which preserve the invariant inner product. The exterior coalgebra of a\nvector space is dual to the symmetric coalgebra of the parity reversion of the\nspace, while the tensor coalgebra of a vector space is dual to the tensor\ncoalgebra of its parity reversion. Using this duality, we introduce a modified\nbracket in the space of coderivations of the tensor and exterior coalgebras\nwhich makes it possible to treat the cohomology of an A-infinity or L-infinity\nas a differential graded algebra in the same manner in which the Gerstenhaber\nbracket is used to transform the Hochschild cochains of an associative algebra\ninto a differential graded algebra.",
"arxiv_id": "q-alg/9512014",
"authors": [
"Michael Penkava"
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"title": "L-infinity algebras and their cohomology",
"url": "https://arxiv.org/abs/q-alg/9512014"
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