We develop the deformation theory of A∞ algebras together with∞-inner products and identify a differential graded Lie algebra that controls the theory. This generalizes the deformation theories of associative algebras, A∞ algebras, associative algebras with inner products, and A∞ algebras with inner products.

We introduce two K-theories, one for vector bundles whose fibers are modules of vertex operator algebras, another for vector bundles whose fibers are modules of associative algebras. We verify the cohomological properties of these K-theories, and construct a natural homomorphism from the VOA K-theory to the associative algebra K-theory.

We present a new approach to cyclic homology that does not involve Connes’ differential and is based on (Ω q A)[u], d + u · ı∆, a noncommutative equivariant de Rham complex of an associative algebra A. Here d is the Karoubi-de Rham differential, which replaces the Connes differential, and ı∆ is an operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit ...

3 Abstract Algebra 32 3.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 32 3.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 32 3.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 32 3.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 The General Associative Law . . . . . . . . . . . . . . . . . . 34 3.6 Identity el...

4 Abstract Algebra 56 4.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 56 4.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 56 4.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 56 4.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 The General Associative Law . . . . . . . . . . . . . . . . . . 58 4.6 Identity el...

3 Abstract Algebra 33 3.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 33 3.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 33 3.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 33 3.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 The General Associative Law . . . . . . . . . . . . . . . . . . 35 3.6 Identity el...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G|+ 1, where |G| is the order of the commutator subgroup. The authors have previously determined the groups G for which this index is maximal and here they determine the G for which it is ‘almost maximal’, that ...

In this paper, we introduce the cohomology theory of O -operators on Hom-associative algebras. This can also be viewed as Hochschild a certain algebra with coefficients in suitable bimodule. Next, study infinitesimal and formal deformations an -operator show that they are governed by above-defined cohomology. Furthermore, notion Nijenhuis elements associated is introduced to characterize trivia...

That it is a Lie homomorphism is precisely the statement of the Jacobi identity. Another exact restatement of the Jacobi identity is contained in the fact that ad takes values in the subalgebra Der(g) of derivations. The notion of a Lie algebra is meant to be an abstraction of the additive commutators of associative algebras. While lie algebras are almost never associative as algebras, they hav...