Suppose that a group G acts by automorphisms on a (restricted) Lie algebra L over a field K of positive characteristic. This gives rise to smash products U(L)#K[G] and u(L)#K[G] . We find necessary and sufficient conditions for these smash products to satisfy a nontrivial polynomial identity.

The Abel identity is (x + y) = n ∑ i=0 ( n i ) x(x − iz)i−1(y + iz)n−i, where x, y and z are real numbers. In this paper we deduce several polynomials expansions, referred to as Abel-type identities, by using Foata’s method, and also show some of their applications.

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 oper...

In our previous papers, the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials and the Casoratian identities for the Askey-Wilson polynomial and its reduced form polynomials were presented. These identities are naturally derived through quantum mechanical formulation of the classical orthogonal polynomials; ordinary quantum mechanics for the former and discrete quantum mechan...

In this paper we study the lower central series {Li}i≥1 of algebras with polynomial identities. More specifically, we investigate the properties of the quotients Ni = Mi/Mi+1 of successive ideals generated by the elements Li. We give a complete description of the structure of these quotients for the free metabelian associative algebra A/(A[A,A][A,A]). With methods from Polynomial Identities the...