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.

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 On some infinitely presented associative algebras

1973 ◽  
Vol 16 (3) ◽  
pp. 290-293 ◽  
Author(s):  
Jacques Lewin
Keyword(s):  
Associative Algebra ◽  
Commutative Algebra ◽  
Polynomial Identity ◽  
Finite Dimension ◽  
Negative Answer ◽  
Free Associative Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutator Ideal ◽  
Finitely Presented

We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.

Some finitely generated associative algebras with a Lie nilpotency identity

2020 ◽  
pp. 2150112
Author(s):  
Vita Glizburg ◽  
Sergey Pchelintsev
Keyword(s):  
Associative Algebra ◽  
Finite Rank ◽  
Alternative Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
The Ideal

It is proved that the algebra of multiplications of the free commutative alternative algebra of finite rank [Formula: see text] is strongly Lie nilpotent of class [Formula: see text]. It is found the class of nilpotency of the ideal, generated by commutators in the free [Formula: see text]-generated associative algebra with identity of Lie nilpotency of degree [Formula: see text] under the condition that [Formula: see text] or [Formula: see text].

scholarly journals AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA

2007 ◽  
Vol 17 (05n06) ◽  
pp. 923-939 ◽  
Author(s):  
A. BELOV-KANEL ◽  
A. BERZINS ◽  
R. LIPYANSKI
Keyword(s):  
Automorphism Group ◽  
Associative Algebra ◽  
Free Associative Algebra ◽  
Free Algebras ◽  
Endomorphism Semigroup ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Group Of Automorphisms ◽  
Variety Of Associative Algebras

Let [Formula: see text] be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety [Formula: see text] freely generated by a set X = {x1,…, xn}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut [Formula: see text], where [Formula: see text] is the subcategory of finitely generated free algebras of the variety [Formula: see text]. The later result solves Problem 3.9 formulated in [17].

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 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.

scholarly journals Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras

2018 ◽  
Vol 25 (3) ◽  
pp. 451-459
Author(s):  
Huishi Li
Keyword(s):  
Free Algebra ◽  
Standard Basis ◽  
Free Algebras ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Generating Set ◽  
Standard Bases ◽  
Generating Sets ◽  
Monomial Ordering ◽  
Minimal Standard

AbstractLet {K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle} be the free algebra generated by {X=\{X_{1},\ldots,X_{n}\}} over a field K. It is shown that, with respect to any weighted {\mathbb{N}}-gradation attached to {K\langle X\rangle}, minimal homogeneous generating sets for finitely generated graded two-sided ideals of {K\langle X\rangle} can be algorithmically computed, and that if an ungraded two-sided ideal I of {K\langle X\rangle} has a finite Gröbner basis {{\mathcal{G}}} with respect to a graded monomial ordering on {K\langle X\rangle}, then a minimal standard basis for I can be computed via computing a minimal homogeneous generating set of the associated graded ideal {\langle\mathbf{LH}(I)\rangle}.

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.

scholarly journals THE HOCHSCHILD COHOMOLOGY RING MODULO NILPOTENCE OF A MONOMIAL ALGEBRA

2006 ◽  
Vol 05 (02) ◽  
pp. 153-192 ◽  
Author(s):  
EDWARD L. GREEN ◽  
NICOLE SNASHALL ◽  
ØYVIND SOLBERG
Keyword(s):  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Finite Dimensional Algebra ◽  
Finitely Generated ◽  
Dimensional Algebra ◽  
Monomial Algebra ◽  
Hochschild Cohomology Ring ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
The Ideal

For a finite dimensional monomial algebra Λ over a field K we show that the Hochschild cohomology ring of Λ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated K-algebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field in [13].

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