Symmetrization of Jordan dialgebras

  title={Symmetrization of Jordan dialgebras},
  author={Murray R. Bremner},
  journal={Nonassociative Mathematics and its
  • M. Bremner
  • Published 31 January 2018
  • Mathematics
  • Nonassociative Mathematics and its Applications
We use computational linear algebra to show that every polynomial identity of degree $n \le 5$ satisfied by the symmetrized Jordan diproduct in every diassociative algebra is a consequence of commutativity. We determine a set of generators for the polynomial identities in degree 6 which do not follow from commutativity. We use the representation theory of the symmetric group to show that there exist further new identities in degree 7. 

Figures from this paper


Special Identities for Quasi-Jordan Algebras
Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities These identities are satisfied by the product ab = a ⊣ b + b ⊢ a in an associativeExpand
On the Definition of Quasi-Jordan Algebra
Velásquez and Felipe recently introduced quasi-Jordan algebras based on the product in an associative dialgebra with operations ⊣ and ⊢. We determine the polynomial identities of degree ≤4 satisfiedExpand
Special and exceptional Jordan dialgebras
In this paper, we study the class of Jordan dialgebras. We develop an approach for reducing problems on dialgebras to the case of ordinary algebras. It is shown that straightforward generalizationsExpand
GröBner-Shirshov Bases for Dialgebras
The Grobner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra are given. Expand
An application of lattice basis reduction to polynomial identities for algebraic structures
Abstract The authors’ recent classification of trilinear operations includes, among other cases, a fourth family of operations with parameter q ∈ Q ∪ { ∞ } , and weakly commutative and weaklyExpand
Varieties of dialgebras and conformal algebras
We introduce and study the concept of a variety of dialgebras which is closely related to the concept of a variety of conformal algebras: Each dialgebra of a given variety embeds into an appropriateExpand
Quasi-Jordan Algebras
In this article we introduce a new algebraic structure of Jordan type and we show several examples. This new structure, called “quasi-Jordan algebras,” appears in the study of the product where x, yExpand
Algebras with one operation including Poisson and other Lie-admissible algebras
Abstract We investigate algebras with one operation. We study when these algebras form a monoidal category and analyze Koszulness and cyclicity of the corresponding operads. We also introduce a newExpand
Universal enveloping algebras of Leibniz algebras and (co)homology
The homology of Lie algebras is closely related to the cyclic homology of associative algebras [LQ]. In [L] the first author constructed a "noncommutative" analog of Lie algebra homology which is,Expand
Un Theoreme de Poincaré–Birkhoff–Witt pour les Algebres de Leibniz
Abstract Poincaré–Birkhoff–Witt's theorem for Leibniz algebras. The dialgebras have been recently introduced by J. L. Loday. For any Leibniz algebra is defined an enveloping dialgebra which has aExpand