Distinguishing simple algebras by means of polynomial identities

Abstract

Our main goal is to extend one of classical Razmyslov's Theorem saying that any two simple finite-dimensional-algebras over an algebraically closed field, satisfying the same polynomial identities, are isomorphic. We suggest a method that allows one to reduce problems about identities of algebras with additional structure to the identities of-algebras. For the convenience of the reader, we start with a full detailed proof of Razmyslov's Theorem. Then we describe our method and its consequences for the identities of graded algebras, algebras with involution, and several others. Keywords Graded algebra • Polynomial identity • Universal algebra Mathematics Subject Classification 17A42 • 08B20 • 16R50 1 Introduction: the problem and some cases In this manuscript we consider the following problem. Given two algebras A and B over the same field, suppose that A and B satisfy the same polynomial identities, is it true that A isomorphic to B? Naturally, this question stated in all its generality has counterexamples. Dedicated to Professor Ivan Shestakov on his 70th birthday.

Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

“…Again , defined by and , is an injective Lie homomorphism, since . By the way, while still holding 2 0, it is no longer true that 2 , 0 and hold. These facts alter the computational behaviour of the L-action on because the generators of the kernel change.…”

“…Therefore and generate a Lie subalgebra of the (Lie) algebra ofendomorphisms of isomorphic to . While the relation suffices to establish a Lie isomorphism between and the Lie subalgebra of generated by the nonzero derivations and , the other relations, namely 2 , 2 0 and (which, together with , implies 0) affect the concrete action of L over : it turns out that the relations 2 2 generate the kernel K of this action. Therefore the L-module structure of is equivalent to the (right) module structure afforded by the factor algebra L K spanned by 1 and .…”

“…Even when no straight Hopf-algebra action is possible, such as for instance for -identities, one can consider a generalized Hopf-algebra action to keep on track ( [4], Remark at page 878. See also [2], Section 7). Of course this does not mean that differences cease to exist: as always happens, general theories are built upon concrete situations, so differential identities, as well as other types of identities, still maintain their characteristic features when dealing with specific problems.…”

“…In the former we regain the results of [11] as byproduct of our general description. On the other hand, 2 is too small to depict the effective impact of the -action on the differential polynomial identities in general cases. All its characteristic features are instead already present when 3, so this case is worth studying to get a more concrete idea of the general one.…”

Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

“…Due to the results in [7] we can associate to every group grading a certain signature. We recall the definition of a signature.…”