  An A-infinity algebra is given by a codifferential on the tensor coalgebra of
a (graded) vector space. An associative algebra is a special case of an
A-infinity algebra, determined by a quadratic codifferential. The notions of
Hochschild and cyclic cohomology generalize from associative to A-infinity
algebras, and classify the infinitesimal deformations of the algebra, and those
deformations preserving an invariant inner product, respectively. Similarly, an
L-infinity algebra is given by a codifferential on the exterior coalgebra of a
vector space, with Lie algebras being special cases given by quadratic
codifferentials. There are natural definitions of cohomology and cyclic
cohomology, generalizing the usual Lie algebra cohomology and cyclic
cohomology, which classify deformations of the algebra and those which preserve
an invariant inner product. This article explores the definitions of these
infinity algebras, their cohomology and cyclic cohomology, and the relation to
their infinitesimal deformations.
