Let A be one of the following commutative associative algebras: the algebra of all smooth functions on the plane: A = C(R), or the algebra of polynomials A = C[p, q] over R or C. There exists a non-trivial formal associative deformation of A called the Moyal ⋆-product (or the standard ⋆-product). It is defined as an associative operation A → A[[h̄]] where h̄ is a formal variable. The explicit for...

The Temperley-Lieb algebra T n with parameter 2 is the associative algebra

Generalizations of the series exp and log to noncommutative nonassociative and other types of algebras were regarded by M. Lazard, and recently by V. Drensky and L. Gerritzen. There is a unique power series exp(x) in one non-associative variable x such that exp(x) exp(x) = exp(2x), exp(0) = 1. We call the unique series H = H(x, y) in two non-associative variables satisfying exp(H) = exp(x) exp(...

In one of our recent papers, the associative and the Lie algebras of Weyl type A[D] = A⊗IF [D] were defined and studied, where A is a commutative associative algebra with an identity element over a field IF of any characteristic, and IF [D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, a class of the above associative and Lie algebras A[D] with I...

We construct an associative algebra with a decomposition into a direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural skew-symmetric bilinear form is a Connes 2-cocycle. Both of them are equivalent to a kind of bialgebra, namely, associative bialgebra...

Given an associative algebra satisfying the left commutativity identity abc = bac (Perm-algebra) with a derivation d, new operation a°b=ad(b) is left-symmetric (pre-Lie). We derive necessary and sufficient conditions for to be embeddable into differential Perm-algebra.