Commutative non-power-associative algebras

2019 ◽  
Vol 29 (08) ◽  
pp. 1527-1539
Author(s):  
Manuel Arenas ◽  
Irvin Roy Hentzel ◽  
Alicia Labra
Keyword(s):  
Associative Algebra ◽  
Associative Algebras ◽  
Commutative Algebras ◽  
Dimension Formula

We study commutative algebras satisfying the identity [Formula: see text] It is known that for [Formula: see text] and for characteristic not [Formula: see text] or [Formula: see text], the algebra is a commutative power-associative algebra. These algebras have been widely studied by Albert, Gerstenhaber and Schafer. For [Formula: see text] Guzzo and Behn in 2014 proved that commutative algebras of dimension [Formula: see text] satisfying [Formula: see text] are solvable. We consider the remaining values of [Formula: see text] We prove that commutative algebras satisfying [Formula: see text] with [Formula: see text] and generated by one element are nilpotent of nilindex [Formula: see text] (we assume characteristic of the field [Formula: see text]).

Quantum Homomorphisms of Associative Algebras

1997 ◽  
Vol 12 (38) ◽  
pp. 2963-2974
Author(s):  
A. E. F. Djemai
Keyword(s):  
Quantum Group ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Type Formula ◽  
Algebra Of Functions ◽  
Inhomogeneous Quantum

Given an associative algebra A generated by {ek, k=1, 2,…} and with an internal law of type: [Formula: see text], we first show that it is possible to construct a quantum bi-algebra [Formula: see text] with unit and generated by (non-necessarily commutative) elements [Formula: see text] satisfying the relations: [Formula: see text]. This leads one to define a quantum homomorphism[Formula: see text]. We then treat the example of the algebra of functions on a set of N elements and we show, for the case N=2, that the resulting bihyphen;algebra is an inhomogeneous quantum group. We think that this method can be used to construct quantum inhomogeneous groups.

Cayley Symmetries in Associative Algebras

1963 ◽  
Vol 15 ◽  
pp. 285-290 ◽  
Author(s):  
Earl J. Taft
Keyword(s):  
Associative Algebra ◽  
Associative Algebras ◽  
Wedderburn Decomposition ◽  
Finite Dimensional ◽  
Principal Theorem ◽  
Algebra Over A Field

Let A be a finite-dimensional associative algebra over a field F. Let R denote the radical of A. Assume that A/R is separable. Then it is well known (the Wedderburn principal theorem) that A possesses a Wedderburn decomposition A = S + R (semi-direct), where S is a separable subalgebra isomorphic with A/R. We call S a Wedderburn factor of A.

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.

Weyl Type Non-Associative Algebras Using Additive Groups I

2007 ◽  
Vol 14 (03) ◽  
pp. 479-488 ◽  
Author(s):  
Seul Hee Choi ◽  
Ki-Bong Nam
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Simple Algebra ◽  
Associative Algebras

A Weyl type algebra is defined in the book [4]. A Weyl type non-associative algebra [Formula: see text] and its restricted subalgebra [Formula: see text] are defined in various papers (see [1, 3, 11, 12]). Several authors find all the derivations of an associative (a Lie, a non-associative) algebra (see [1, 2, 4, 6, 11, 12]). We define the non-associative simple algebra [Formula: see text] and the semi-Lie algebra [Formula: see text], where [Formula: see text]. We prove that the algebra is simple and find all its non-associative algebra derivations.

Cleft extensions and Galois extensions for Hom-associative algebras

2016 ◽  
Vol 27 (03) ◽  
pp. 1650025 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Associative Algebra ◽  
Galois Extension ◽  
Classical Case ◽  
Normal Basis ◽  
Associative Algebras ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions

In this paper, we consider Hom-(co)modules associated to a Hom-(co)associative algebra and define the notion of Hom-triple. We introduce the definitions of cleft extension and Galois extension with normal basis in this setting and we show that, as in the classical case, these notions are equivalent in the Hom setting.

scholarly journals Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

2014 ◽  
Vol 24 (08) ◽  
pp. 1157-1182 ◽  
Author(s):  
Roberto La Scala
Keyword(s):  
Associative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Polynomial Rings ◽  
Free Associative Algebra ◽  
Commutative Algebras ◽  
Skew Polynomial Rings ◽  
Experimental Implementation ◽  
Buchberger Algorithm ◽  
Skew Polynomial

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

Identities of Non-Associative Algebras

1965 ◽  
Vol 17 ◽  
pp. 78-92 ◽  
Author(s):  
J. Marshall Osborn
Keyword(s):  
Sufficient Conditions ◽  
Interesting Property ◽  
Necessary And Sufficient Conditions ◽  
Partial Ordering ◽  
Associative Algebras ◽  
Commutative Algebras ◽  
Odd Degree ◽  
Necessary And Sufficient

In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper.

scholarly journals Some degeneracy and pathology in non-associative radical theory II

1981 ◽  
Vol 23 (3) ◽  
pp. 423-428 ◽  
Author(s):  
B. J. Gardner
Keyword(s):  
Associative Algebras ◽  
Additive Structure ◽  
Radical Theory ◽  
Commutative Algebras ◽  
Unital Ring ◽  
Universal Classes

It is shown that in the universal classes of(i) all commutative algebras,(ii) all anti-commutative algebras and(iii) all algebras satisfying x2 = 0 (over any commutative, associative, unital ring)the only radical classes with hereditary semi-simple classes are those for which membership is determined by additive structure. Some examples of non-hereditary semi-simple classes in the class of all power-associative algebras are also presented.

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 Carter subgroups in the group of units of an associative algebra

2005 ◽  
Vol 71 (3) ◽  
pp. 471-478 ◽  
Author(s):  
Thorsten Bauer ◽  
Salvatore Siciliano
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Group Of Units ◽  
Cartan Subalgebras

In this paper we examine some properties of the Carter subgroups in the group of units of certain associative algebras. A description of the Carter subgroups in the case of a solvable associative algebra is obtained. Moreover, given an associative algebra A, we study relationships between the Cartan subalgebras of the Lie algebra associated with A and the Carter subgroups of the group of units of A.

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