WEYL TYPE ALGEBRAS FROM QUANTUM TORI

2006 ◽  
Vol 08 (02) ◽  
pp. 135-165 ◽  
Author(s):  
KAIMING ZHAO
Keyword(s):  
Differential Operator ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Operator Algebra ◽  
Laurent Polynomials ◽  
Associative Algebras ◽  
Defining Relations ◽  
Quantum Tori ◽  
Quantum Version ◽  
Algebra Over A Field

We introduce and study the quantum version of the differential operator algebra on Laurent polynomials and its associated Lie algebra over a field F of characteristic 0. The q-quantum torus Fq is the unital associative algebra over F generated by [Formula: see text] subject to the defining relations titj = qi,jtjti, where qi,i = 1, [Formula: see text]. Let D be a subspace of [Formula: see text] where ∂i is the derivation on Fq sending [Formula: see text] to [Formula: see text]. Then, the quantum differential operator algebra is the associative algebra Fq[D]. Assume that Fq[D] is simple as an associative algebra. We compute explicitly all 2-cocycles of Fq[D], viewed as a Lie algebra. More precisely, we show that the second cohomology group of Fq[D] has dimension n if D = 0, dimension 1 if dim D = 1, and dimension 0 if dim D > 1. We also determine all isomorphisms and anti-isomorphisms Fq[D] → Fq′[D′] of simple associative algebras, and all isomorphisms Fq[D]/F → Fq′[D′]/F of simple Lie algebras.

Growth functions of lie algebras associated with associative algebras

2021 ◽  
pp. 2250110
Author(s):  
Adel Alahmadi ◽  
Fawziah Alharthi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Finite Collection ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Growth Functions ◽  
Nilpotent Elements ◽  
Algebra Over A Field

Let [Formula: see text] be a finitely generated associative algebra over a field [Formula: see text] of characteristic [Formula: see text] and let [Formula: see text] be its associated Lie algebra. In this paper, we investigate relations between the growth functions of [Formula: see text] and the Lie algebra [Formula: see text]. We prove that if A is generated by a finite collection of nilpotent elements, then the growth functions are asymptotically equivalent.

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.

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

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.

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.

Standard Gröbner-Shirshov Bases of Free Algebras Over Rings, I

1998 ◽  
Vol 08 (06) ◽  
pp. 689-726 ◽  
Author(s):  
Alexander A. Mikhalev ◽  
Andrej A. Zolotykh
Keyword(s):  
Associative Algebra ◽  
Standard Basis ◽  
Free Associative Algebra ◽  
Semigroup Algebras ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Standard Bases ◽  
Equality Problem ◽  
Algebra Over A Field ◽  
The Ideal

We consider standard bases of ideals of free associative algebras over rings. The main result of the article is a criterion for a subset of a free associative algebra to be a standard basis of the ideal it generates. Based on this result, we present an infinite algorithm to construct the reduced standard basis of an ideal. A generalization in case of some semigroup algebras is presented. We also describe a way to construct weak standard bases and reduced standard bases of ideals of a free associative algebra over an arbitrary finitely generated ring (over a finitely generated algebra over a field). Some examples of constructions of standard bases and of solutions of the equality problem are included.

Power-Associative Algebras in which Every Subalgebra is a Left Ideal

1970 ◽  
Vol 13 (2) ◽  
pp. 239-243
Author(s):  
D. J. Rodabaugh
Keyword(s):  
Left Ideal ◽  
Associative Algebra ◽  
Nonassociative Algebra ◽  
Single Element ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
Algebra Over A Field

By an L-algebra we mean a power-associative nonassociative algebra (not necessarily finite-dimensional) over a field F in which every subalgebra generated by a single element is a left ideal. An H-algebra is a power-associative algebra in which every subalgebra is an ideal. The H-algebras were characterized by D. L. Outcalt in [2]. Let Sα be the semigroup with cardinality α such that if x, y ∊ Sα then xy = y. Consider the algebra over a field F with basis Sα. Such an algebra is an L-algebra that is not an H-algebra unless Sα contains only one element. In this paper we will prove that an algebra A over a field F with char. ≠ 2 is an L-algebra if and only if it is either an H-algebra or has a basis Sα where α is the dimension of A.

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.

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