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 Decompositions of algebras and post-associative algebra structures

2019 ◽  
Vol 30 (03) ◽  
pp. 451-466
Author(s):  
Dietrich Burde ◽  
Vsevolod Gubarev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Algebra Structure ◽  
Associative Algebras ◽  
Simple Lie Algebra ◽  
Reductive Lie Algebra ◽  
Baxter Operator

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

scholarly journals Jordan algebras arising in population genetics

1967 ◽  
Vol 15 (4) ◽  
pp. 291-294 ◽  
Author(s):  
P. Holgate
Keyword(s):  
Population Genetics ◽  
Associative Algebra ◽  
Jordan Algebras ◽  
Associative Algebras

The non-associative algebras arising in genetics (1), are rather isolated from other branches of non-associative algebra (6). However, in a paper (5), in which he studied these algebras in terms of their transformation algebras, Schafer proved that the gametic and zygotic algebras for a single diploid locus are Jordan algebras.

COTRIPLE HOMOLOGY OF CROSSED 2-CUBES OF LIE ALGEBRAS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350007
Author(s):  
G. DONADZE ◽  
M. LADRA
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Low Dimensional ◽  
The Relationship ◽  
Natural Way

The cotriple homology of crossed 2-cubes of Lie algebras is constructed and investigated. Namely, we calculate the cotriple homology of an inclusion crossed 2-cube of Lie algebras in terms of the bi-relative Chevalley–Eilenberg homologies. We also define in a natural way the Chevalley–Eilenberg homology of crossed 2-cubes of Lie algebras and study the relationship between cotriple and Chevalley–Eilenberg homologies for any crossed 2-cube of Lie algebras. We show that low-dimensional cyclic homologies of associative algebras are calculated in terms of the cotriple homology of crossed 2-cubes of Lie algebras.

scholarly journals Co-stability of Radicals and Its Applications to PI-Theory

2016 ◽  
Vol 23 (03) ◽  
pp. 481-492 ◽  
Author(s):  
A. S. Gordienko
Keyword(s):  
Hopf Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Jacobson Radical ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
Comodule Algebra ◽  
Graded Polynomial Identities ◽  
Finite Dimensional Lie Algebras ◽  
Algebra Over A Field

We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.

scholarly journals Finite generation of Lie derived powers of associative algebras

2019 ◽  
Vol 18 (03) ◽  
pp. 1950059
Author(s):  
Adel Alahmadi ◽  
Hamed Alsulami
Keyword(s):  
Lie Algebras ◽  
Associative Algebra ◽  
Finite Collection ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Finite Generation ◽  
Nilpotent Elements ◽  
Algebra Over A Field

Let [Formula: see text] be an associative algebra over a field of characteristic [Formula: see text] that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of [Formula: see text] are finitely generated Lie 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 GRÖBNER–SHIRSHOV BASES AND EMBEDDINGS OF ALGEBRAS

2010 ◽  
Vol 20 (07) ◽  
pp. 875-900 ◽  
Author(s):  
L. A. BOKUT ◽  
YUQUN CHEN ◽  
QIUHUI MO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Differential Algebras

In this paper, by using Gröbner–Shirshov bases, we show that in the following classes, each (respectively, countably generated) algebra can be embedded into a simple (respectively, two-generated) algebra: associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We show that in the following classes, each countably generated algebra over a countable field k can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative Ω-algebras, associative λ-differential algebras. We give another proofs of the well known theorems: each countably generated group (respectively, associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (respectively, associative algebra, semigroup, Lie 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]).

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