Linfinity algebras and their cohomology
Abstract
An associative algebra is nothing but an odd quadratic codifferential on the tensor coalgebra of a vector space, and an Ainfinity algebra is simply an arbitrary odd codifferential. Hochschild cohomology classifies the deformations of an associative algebra into an Ainfinity algebra, and cyclic cohomology in the presence of an invariant inner product classifies the deformations of the associative algebra into an Ainfinity 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 Linfinity algebra is a more general codifferential. In this paper, ordinary and cyclic cohomology of Linfinity 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 Linfinity algebra. Similarly, the cyclic cohomology of a Lie algebra with an invariant inner product classifies the deformations of the Lie algebra into an Linfinity 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 Ainfinity or Linfinity 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.
 Publication:

eprint arXiv:qalg/951201
 Pub Date:
 December 1995
 arXiv:
 arXiv:qalg/9512014
 Bibcode:
 1995q.alg....12014P
 Keywords:

 Mathematics  Quantum Algebra
 EPrint:
 28 pages, amslatex document