We consider the lower central filtration of the free associative algebra An with n generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebraWn of polynomial vector fields on C. We compute the space [An, An]/[An, [An, An...

In the theory of dynamical Yang-Baxter equation, with any Hopf algebra H and a certain H-module and H-comodule algebra L (base algebra) one associates a monoidal category. Given an algebra A in that category, one can construct an associative algebra A⋊L, which is a generalization of the ordinary smash product when A is an ordinary H-algebra. We study this ”dynamical smash product” and its modul...

Let Ä<2> = R that keeps (xy — yx) fixed (up to multiplication by an element of R), then . This follows from a more general result about endomorphisms of Ä...

The goal of the paper is twofold: it aims to give an extensive set tools and bibliography towards Nowicki's conjecture both in for polynomial algebras associative setting; establishes a new result about free metabelian Poisson algebra.

We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finte-dimensional, connected gradation. Dually, the vector space R〈x0, x1〉 is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H, and an associative, commutative algebra for the shuffle product. These two structures admit a compatibilit...

As Don Schack mentioned in his plenary talk [SDS], Gerstenhaber [MG] observed that if A is an associative algebra, then the Hochschild cohomology of A is a graded commutative algebra with an additional structure, viz., that of a Lie algebra. The two structures satisfy a compatibility condition (graded Poission structure). We agree to call such structures G–algebras (Gerstenhaber–algebras). Scha...

A class of H-analytic (differentiable by Hausdorff) functions in a three-dimensional noncommutative algebra e $$ {\overset{\sim }{\mathbbm{A}}}_2 over the field ℂ is considered. All are described explicit form. The obtained description applied to integral representation these functions, and mentioned also when solving some PDEs.

The substitution box (S-box) is a fundamentally important component of symmetric key cryptosystem. An S-box primary source non-linearity in modern block ciphers, and it resists the linear attack. Various approaches have been adopted to construct S-boxes. S-boxes are commonly constructed over commutative associative algebraic structures including Galois fields, unitary rings cyclic non-cyclic fi...

In 2013 Benkart, Lopes and Ondrus introduced studied in a series of papers the infinite-dimensional unital associative algebra Ah generated by elements x,y, which satisfy relation yx?xy=h for some 0?h?F[x]. We generalize this construction to Ah(B) working over fixed F-algebra B instead F. describe polynomial identities infinite field F case h?B[x] satisfies certain restrictions.