No abstract

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm introduced by Bremner and Sánchez-Ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra. We apply the BSO algorithm to the Jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the KP algorithm; this provides a large class of examples of Jordan triple disystems. We formulate a general conjecture expressed by a commutative diagram relating the output of the KP and BSO algorithms. We conclude by generalizing the Jordan triple product in a Jordan algebra to operations in a Jordan dialgebra; we use computer algebra to verify that resulting structures provide further examples of Jordan triple disystems. For this last result, we also provide an independent theoretical proof using Jordan structure theory.

Communicated by S. R. López-PermouthThe notions of annihilator ideal and split structure are studied in detail for both, dialgebras and quasi-Jordan algebras. It yields methods for additional units in the two structures. As a consequence the notion of regular element receives special attention. . Downloaded from www.worldscientific.com by PENNSYLVANIA STATE UNIVERSITY on 03/15/15. For personal use only.If the bracket is skew-symmetric, then L is a Lie algebra. Therefore Lie algebras are particular cases of Leibniz algebras. Example 2. Let L be a Lie algebra and let M be a L-module with action M ×L → M , (m, x) → mx. Let f : M → L be a L-equivariant linear map, this is f (mx) = [f (m), x], for all m ∈ M and x ∈ L, then one can put a Leibniz structure on M as follows [m, n] := mf (n), for all m, n ∈ M. Additionally, the map f defines a homomorphism between Leibniz algebras, since f ([m, n] ) = f (mf (n)) = [f (m), f(n)].Example 3. Let (A, d) be a differential associative algebra. So, by hypothesis, d(ab) = da b + a db and d 2 = 0. We define the bracket on A by the formula [a, b] := a db − db a, then the vector space A equipped with this bracket is a Leibniz algebra. J. Algebra Appl. 2009.08:191-218. Downloaded from www.worldscientific.com by PENNSYLVANIA STATE UNIVERSITY on 03/15/15. For personal use only.Split Dialgebras,