THE TRANSFER OF A COMMUTATOR IDENTITY IN A NIL-GENERATED ALGEBRA

2002 ◽  
Vol 12 (03) ◽  
pp. 437-443 ◽  
Author(s):  
A. N. KRASILNIKOV ◽  
D. M. RILEY
Keyword(s):  
Associative Algebra ◽  
Adjoint Group ◽  
Commutator Identity ◽  
Nilpotent Elements ◽  
Algebra Over A Field

We show that if an associative algebra over a field of characteristic 0 is generated by its nilpotent elements and satisfies a multilinear Lie commutator identity then its adjoint group satisfies the corresponding multilinear group commutator identity.

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.

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.

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.

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.

The Second Cohomology of Current Algebras of General Lie Algebras

2008 ◽  
Vol 60 (4) ◽  
pp. 892-922 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Friedrich Wagemann
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Characteristic Zero ◽  
Cohomology Space ◽  
Trivial Module ◽  
Commutative Associative Algebra ◽  
Locally Convex ◽  
Algebra Over A Field

AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

Some Division Algebras

1959 ◽  
Vol 11 ◽  
pp. 51-58
Author(s):  
Joseph L. Zemmer
Keyword(s):  
Linear Space ◽  
Associative Algebra ◽  
Bilinear Form ◽  
Division Algebras ◽  
Image Position ◽  
Algebra Over A Field

Let K* be an associative algebra over a field F with identity u, and let u, e1, e2, … , be a basis for K*. Denote by K the linear space, over F, spanned by the ei,i = 1, 2, … . Then for x, y in K, xy = αu + a, where a ∈ K. Define h(x, y) = α and x.y = a. With respect to the operation thus defined, K becomes an algebra over F satisfying1Further, the bilinear form h(x, y) is associative on K. Any algebra, over a field F, which possesses an associative bilinear form h(x, y) and satisfies (1) will be called a algebra. It is not difficult to show that any algebra K can be obtained from a unique associative algebra K* with identity by the process described above. The algebra K* will be called the associated associative algebra of K.

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