QUANDLE G-ALGEBRA OVER A FIELD

2020 ◽  
Vol 48 (1) ◽  
pp. 83-108
Author(s):  
Ahmad M. Alghamdi ◽  
Amani M. Alfadhli
Keyword(s):  
Algebra Over A Field

scholarly journals NORMAL DOMAINS WITH MONOMIAL PRESENTATIONS

2009 ◽  
Vol 19 (03) ◽  
pp. 287-303 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Class Group ◽  
Semigroup Algebra ◽  
Closed Domain ◽  
Finitely Generated ◽  
Defining Relations ◽  
Integrally Closed ◽  
Algebra Over A Field

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X1,…, Xn | R〉, where R is a set of monomial relations in the generators X1,…, Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1,…, Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

scholarly journals Duality and socle generators for residual intersections

2019 ◽  
Vol 2019 (756) ◽  
pp. 183-226 ◽  
Author(s):  
David Eisenbud ◽  
Bernd Ulrich
Keyword(s):  
Gorenstein Ring ◽  
Jacobian Determinant ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Duality Results ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Residue Theory

AbstractWe prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt.Suppose that I is an ideal of codimension g in a Gorenstein ring, and {J\subset I} is an ideal with {s=g+t} generators such that {K:=J:I} has codimension s. Let {{\overline{I}}} be the image of I in {{\overline{R}}:=R/K}.In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules {{\overline{I}}{}^{u}} and {{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into {{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}}.In the second part of the paper we study the analogue of residue theory, and prove that, when {R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of {I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.

Necessary conditions for power commuting in finite-dimensional algebras over a field

2018 ◽  
Vol 28 (5) ◽  
pp. 339-344
Author(s):  
Andrey V. Zyazin ◽  
Sergey Yu. Katyshev
Keyword(s):  
Necessary Conditions ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Finite Dimensional Algebras

Abstract Necessary conditions for power commuting in a finite-dimensional algebra over a field are presented.

scholarly journals ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES

2009 ◽  
Vol 86 (1) ◽  
pp. 1-15 ◽  
Author(s):  
JONATHAN BROWN ◽  
JONATHAN BRUNDAN
Keyword(s):  
Lie Algebra ◽  
Characteristic Zero ◽  
Universal Enveloping Algebra ◽  
General Linear ◽  
Enveloping Algebra ◽  
Nilpotent Matrix ◽  
Nilpotent Matrices ◽  
Algebra Over A Field

AbstractWe construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

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.

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

ON SOME EMBEDDING OF LIE ALGEBRAS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250017 ◽  
Author(s):  
E. CHIBRIKOV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Free Generators ◽  
Algebra Over A Field

In this paper we prove that every recursively presented Lie algebra over a field which is finite extention of its simple subfield can be embedded in a recursively presented Lie algebra defined by relations which are equalities of (nonassociative) words of generators and α + β = γ(α, β, γ are free generators).

On the Hopf algebra dual of an enveloping algebra

1982 ◽  
Vol 91 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Stephen Donkin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Groups ◽  
Field Extension ◽  
Smash Product ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Polycyclic Groups ◽  
Algebra Over A Field

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

scholarly journals Identities of the left-symmetric Witt algebras

2016 ◽  
Vol 26 (02) ◽  
pp. 435-450 ◽  
Author(s):  
Daniyar Kozybaev ◽  
Ualbai Umirbaev
Keyword(s):  
Lie Algebras ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Witt Algebra ◽  
Symmetric Algebras ◽  
Algebra Over A Field

Let [Formula: see text] be the polynomial algebra over a field [Formula: see text] of characteristic zero in the variables [Formula: see text] and [Formula: see text] be the left-symmetric Witt algebra of all derivations of [Formula: see text] [D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4(3) (2006) 323–357]. We describe all right operator identities of [Formula: see text] and prove that the set of all algebras [Formula: see text], where [Formula: see text], generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for [Formula: see text].

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