Algebras, dialgebras, and polynomial identities

Abstract

This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc − bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raúl Felipe, Luiz A. Peresi, and Juana Sánchez-Ortega. AlgebrasThroughout this talk the base field F will be arbitrary, but we usually exclude low characteristics, especially p ≤ n where n is the degree of the polynomial identities under consideration. The assumption p > n allows us to assume that all polynomial identities are multilinear and that the group algebra FS n is semisimple.Definition 1.1. An algebra is a vector space A with a bilinear operationUnless otherwise specified, we write ab = µ(a, b) for a, b ∈ A. We say that A is associative if it satisfies the polynomial identity (ab)c ≡ a(bc).Throughout this paper we will use the symbol ≡ to indicate an equation that holds for all values of the arguments; in this case, all a, b, c ∈ A.Theorem 1.2. The free unital associative algebra on a set X of generators has basis consisting of all words of degree n ≥ 0,with the product defined on basis elements by concatenation and extended bilinearly,

Hom-post-Lie modules, O-operators and some functors on Hom-algebras

“…for all x, y, z ∈ A. Recall also that a Hom-algebra (A, •, α) is said to be left commutative [31], if…”