scholarly journals How to compute the Wedderburn decomposition of a finite-dimensional associative algebra

2011 ◽  
Vol 3 (1) ◽  
Author(s):  
Murray R. Bremner
Keyword(s):  
Associative Algebra ◽  
Wedderburn Decomposition ◽  
Finite Dimensional

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 CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

scholarly journals Maximal subalgebras of finite-dimensional algebras

2019 ◽  
Vol 31 (5) ◽  
pp. 1283-1304 ◽  
Author(s):  
Miodrag Cristian Iovanov ◽  
Alexander Harris Sistko
Keyword(s):  
Associative Algebra ◽  
Complete Classification ◽  
Maximal Subalgebras ◽  
Incidence Algebras ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Closed Fields ◽  
Representations Of Algebras ◽  
Algebraically Closed Fields ◽  
Full Classification

AbstractWe study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

Noetherian Banach Jordan pairs

2001 ◽  
Vol 130 (1) ◽  
pp. 25-36 ◽  
Author(s):  
N. BOUDI ◽  
H. MARHNINE ◽  
C. ZARHOUTI ◽  
A. FERNANDEZ LOPEZ ◽  
E. GARCIA RUS
Keyword(s):  
Recent Work ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Jacobson Radical ◽  
Alternative Algebra ◽  
Chain Condition ◽  
Finite Capacity ◽  
Jordan Pair ◽  
Finite Dimensional ◽  
Ascending Chain Condition

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.Here we show the following results:(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

scholarly journals On H-simple not necessarily associative algebras

2019 ◽  
Vol 18 (09) ◽  
pp. 1950162
Author(s):  
A. S. Gordienko
Keyword(s):  
Associative Algebra ◽  
Hopf Algebras ◽  
Simple Algebra ◽  
Polynomial Identities ◽  
Associative Algebras ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Graded Polynomial Identities ◽  
Group Gradings ◽  
Polynomial Formula

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

Classes of Functions on Algebras

1966 ◽  
Vol 18 ◽  
pp. 139-146 ◽  
Author(s):  
C. G. Cullen ◽  
C. A. Hall
Keyword(s):  
Associative Algebra ◽  
Real Field ◽  
Complex Field ◽  
The Real ◽  
Finite Dimensional

Let be a finite-dimensional linear associative algebra over the real field R or the complex field C and let F be a function with domain and range in .Several classes of functions on have been discussed in the literature, and it is the purpose of this paper to discuss the relationships between these classes and to present some interesting examples. First we shall list the definitions of the classes we wish to consider here.

A Note on Algebras that are Sums of Two Subalgebras

2016 ◽  
Vol 59 (2) ◽  
pp. 340-345 ◽  
Author(s):  
Marek Kępczyk
Keyword(s):  
Associative Algebra ◽  
Polynomial Identity ◽  
Arbitrary Field ◽  
Polynomial Identities ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras

AbstractWe study an associative algebra A over an arbitrary field that is a sum of two subalgebras B and C (i.e., A = B+C). We show that if B is a right or left Artinian PI algebra and C is a PI algebra, then A is a PI algebra. Additionally, we generalize this result for semiprime algebras A. Consider the class of all semisimple finite dimensional algebras A = B + C for some subalgebras B and C that satisfy given polynomial identities f = 0 and g = 0, respectively. We prove that all algebras in this class satisfy a common polynomial identity.

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