9 research outputs found
Jordan Triple Disystems
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
Sanchez-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.Comment: 23 page
Algebras, dialgebras, and polynomial identities
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 Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.Comment: 32 page
Derivations and deformations of -Jordan Lie supertriple systems
Let be a -Jordan Lie supertriple system. We first introduce the
notions of generalized derivations and representations of and present some
properties. Also, we study the low dimension cohomology and the coboundary
operator of , and then we investigate the deformations and Nijenhuis
operators of by choosing some suitable cohomology.Comment: 23page
Non-Associative Algebraic Structures: Classification and Structure
These are detailed notes for a lecture on "Non-associative Algebraic
Structures: Classification and Structure" which I presented as a part of my
Agrega\c{c}\~ao em Matem\'atica e Applica\c{c}\~oes (University of Beira
Interior, Covilh\~a, Portugal, 13-14/03/2023)