scholarly journals On Jordan Algebras and Some Unification Results

2020 ◽  
Author(s):  
Florin Nichita
Keyword(s):  
Differential Geometry ◽  
Lie Algebras ◽  
Algebraic Structure ◽  
International Workshop ◽  
Jordan Algebras ◽  
Baxter Equation ◽  
Associative Algebras ◽  
Associative Structures

This paper is based on a talk given at the 14-th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between July 9-th and July 11-th, 2019. After presenting some historical facts, we will consider some geometry problems related to unification approaches. Jordan algebras and Lie algebras are the main non-associative structures. Attempts to unify non-associative algebras and associative algebras led to UJLA structures. Another algebraic structure which unifies non-associative algebras and associative algebras is the Yang-Baxter equation. We will review topics relared to the Yang-Baxter equation and Yang-Baxter systems, with the goal to unify constructions from Differential Geometry.

Baer-invariants and extensions relative to a variety

1967 ◽  
Vol 63 (3) ◽  
pp. 569-578 ◽  
Author(s):  
Abraham S.-T. Lue
Keyword(s):  
Commutative Ring ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Jordan Algebras ◽  
Associative Algebras ◽  
Definition Of ◽  
The Relationship

This paper examines the relationship between extensions in a variety and general extensions in the category of associative algebras. Our associative algebras are all unitary, over some fixed commutative ring Λ with identity, but while our discussion will be restricted to this category, it is clear that obvious analogues exist for groups, Lie algebras and Jordan algebras. (We use the notion of a bimultiplication of an associative algebra. In (2), Knopfmacher gives the definition of a bimultiplication in any variety of linear algebras.)

scholarly journals Unification Theories: Examples and Applications

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 85 ◽  
Author(s):  
Florin Nichita
Keyword(s):  
Lie Algebras ◽  
Jordan Algebras ◽  
Associative Algebras ◽  
Transcendental Numbers

We consider several unification problems in mathematics. We refer to transcendental numbers. Furthermore, we present some ways to unify the main non-associative algebras (Lie algebras and Jordan algebras) and associative algebras.

scholarly journals Pre-jordan Algebras

2013 ◽  
Vol 112 (1) ◽  
pp. 19 ◽  
Author(s):  
Dongping Hou ◽  
Xiang Ni ◽  
Chengming Bai
Keyword(s):  
Lie Algebras ◽  
Jordan Algebra ◽  
Symplectic Form ◽  
Jordan Algebras ◽  
Multiplication Operators ◽  
Baxter Equation ◽  
Algebraic Structures ◽  
Left Multiplication ◽  
Close Relationship ◽  
Dendriform Algebras

The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of $\mathcal{O}$-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre-Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of $\mathcal{O}$-operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.

scholarly journals Unification Theories. Examples and Applications

2018 ◽  
Author(s):  
Florin F. Nichita
Keyword(s):  
Lie Algebras ◽  
Jordan Algebras ◽  
Associative Algebras ◽  
Transcendental Numbers

We consider several unification problems in mathematics. We refer to transcendental numbers. Also, we present some ways to unify the main non-associative algebras (Lie algebras and Jordan algebras) and associative algebras.

Some Lie Admissible Algebras

1962 ◽  
Vol 14 ◽  
pp. 287-292 ◽  
Author(s):  
P. J. Laufer ◽  
M. L. Tomber
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jordan Algebras ◽  
Associative Algebras ◽  
Alternative Algebras

Several studies have been made to obtain larger classes of non-associative algebras from classes of algebras with a known structure. Thus, we have right alternative algebras (2)* and non-commutative Jordan algebras (6), (7), (8), and (9). These algebras are defined by a subset of the set of identities of the algebras from which they derive their names. Also, Albert (1), among others has studied Jordan admissible algebras. This paper is concerned with algebras which are related to Lie algebras in that they satisfy some of the identities of a Lie algebra and are Lie admissible. Theorem 2 answers a question raised by Albert in (1).

Continuity of derivations on Mal'cev H*-algebras

1991 ◽  
Vol 110 (3) ◽  
pp. 455-459 ◽  
Author(s):  
Borut Zalar
Keyword(s):  
Lie Algebras ◽  
Jordan Algebras ◽  
Associative Algebras ◽  
Alternative Algebras ◽  
Mal’Cev Algebras ◽  
Long Time

A long time ago the concept of H*-algebra was introduced by Ambrose in [1] where the structure of complex associative H*-algebras was given. Since then this theory was extended to such classical types of non-associative algebras as alternative algebras (in [6]), Jordan algebras (in [5, 13, 14]), non-commutative Jordan algebras (in [5]), Lie algebras (in [3, 9, 10]) and Mal'cev algebras (in [2]).

scholarly journals Algebraic Structure of the Lorentz and of the Poincaré Lie Algebras

2018 ◽  
Vol 41 (2) ◽  
pp. 305-346
Author(s):  
Pablo ALBERCA BJERREGAARD ◽  
Dolores MARTÍN BARQUERO ◽  
Cándido MARTÍN GONZÁLEZ ◽  
Daouda NDOYE
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure
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