scholarly journals CMMSE: Combinatorial structures and finite-dimensional alternative algebras

2021 ◽  
Author(s):  
Manuel Ceballos
Keyword(s):  
Theoretical Study ◽  
Alternative Algebra ◽  
New Method ◽  
Combinatorial Structure ◽  
Combinatorial Structures ◽  
Isomorphism Classes ◽  
Alternative Algebras ◽  
Finite Dimensional

In this paper, the link between combinatorial structures and alternative algebras is studied, determining which configurations are associated with those algebras. Moreover, the isomorphism classes of each 2-dimensional configuration associated with these algebras is analyzed, providing a new method to classify them. In order to complement the theoretical study, two algorithmic methods are implemented: the first one constructs and draws the (pseudo)digraph associated with a given alternative algebra and the second one tests if a given combinatorial structure is associated with some alternative algebra.

scholarly journals Right alternative algebras with commutators in a nucleus

1992 ◽  
Vol 46 (1) ◽  
pp. 81-90
Author(s):  
Erwin Kleinfeld ◽  
Harry F. Smith
Keyword(s):  
Associative Ring ◽  
Linear Span ◽  
Alternative Algebra ◽  
Wedderburn Decomposition ◽  
Alternative Algebras ◽  
Finite Dimensional ◽  
Characteristic 2 ◽  
The Right ◽  
Nil Radical ◽  
Algebra Over A Field

Let A be a right alternative algebra, and [A, A] be the linear span of all commutators in A. If [A, A] is contained in the left nucleus of A, then left nilpotence implies nilpotence. If [A, A] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [A, A] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A, then A has a Wedderburn decomposition

Finite-dimensional Leibniz algebras and combinatorial structures

2017 ◽  
Vol 20 (01) ◽  
pp. 1750004 ◽  
Author(s):  
M. Ceballos ◽  
J. Núñez ◽  
Á. F. Tenorio
Keyword(s):  
Leibniz Algebra ◽  
Combinatorial Structure ◽  
Combinatorial Structures ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Dimension 2

Given a finite-dimensional Leibniz algebra with certain basis, we show how to associate such algebra with a combinatorial structure of dimension 2. In some particular cases, this structure can be reduced to a digraph or a pseudodigraph. In this paper, we study some theoretical properties about this association and we determine the type of Leibniz algebra associated to each of them.

scholarly journals Invariant connections and invariant holomorphic bundles on homogeneous manifolds

2014 ◽  
Vol 12 (1) ◽  
pp. 1-13
Author(s):  
Indranil Biswas ◽  
Andrei Teleman
Keyword(s):  
Lie Group ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Transitive Action ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Group A ◽  
Holomorphic Bundles ◽  
Invariant Gauge

AbstractLet X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

scholarly journals Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

Kupershmidt-(dual-)Nijenhuis structure on the alternative algebra with a representation

2020 ◽  
pp. 2150097
Author(s):  
Qinxiu Sun
Keyword(s):  
Alternative Algebra ◽  
Alternative Algebras ◽  
Alternative Formula

The aim of this paper is to study Kupershmidt-(dual-)Nijenhuis structures on alternative algebras with representations. The notion of a (dual-)Nijenhuis pair is introduced and it can generate a trivial deformation of an alternative algebra with a representation. We introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on an alternative algebra with a representation. Furthermore, we verify that Kupershmidt operators and Kupershmidt-(dual-)Nijenhuis structures can give rise to each other under some conditions. Finally, we study the notions of Rota–Baxter–Nijenhuis structures and alternative [Formula: see text]-matrix-Nijenhuis structures. Meanwhile, we investigate the relation between them.

scholarly journals Strongly simply connected Auslander algebras

1997 ◽  
Vol 39 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Ibrahim Assem ◽  
Peter Brown
Keyword(s):  
Representation Theory ◽  
Closed Field ◽  
Interesting Problem ◽  
Indecomposable Modules ◽  
Finite Algebras ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Infinite Case ◽  
Simple Connectedness ◽  
Simply Connected

Letkbe an algebraically closed field. By an algebra is meant an associative finite dimensionalk-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

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.

Action theory of alternative algebras

2019 ◽  
Vol 26 (2) ◽  
pp. 177-197
Author(s):  
José Manuel Casas ◽  
Tamar Datuashvili ◽  
Manuel Ladra
Keyword(s):  
Action Theory ◽  
Alternative Algebra ◽  
Alternative Algebras

Abstract We present the category of alternative algebras as a category of interest. This kind of approach enables us to describe derived actions in this category, study their properties and construct a universal strict general actor of any alternative algebra. We apply the results obtained in this direction to investigate the problem of the existence of an actor in the category of alternative algebras.

On Degenerations of Modules With Nondirecting Indecomposable Summands

1996 ◽  
Vol 48 (5) ◽  
pp. 1091-1120 ◽  
Author(s):  
A. Skowroński ◽  
G. Zwara
Keyword(s):  
Natural Number ◽  
Partial Order ◽  
General Linear Group ◽  
Partial Orders ◽  
Connected Components ◽  
Closed Field ◽  
Affine Variety ◽  
Indecomposable Modules ◽  
Isomorphism Classes ◽  
Finite Dimensional

AbstractLet A be a finite dimensional associative K-algebra with an identity over an algebraically closed field K, d a natural number, and modA(d) the affine variety of d-dimensional A-modules. The general linear group Gld(K) acts on modA(d) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules. For M and N in modA(d), N is called a degeneration of M, if TV belongs to the closure of the orbit of M. This defines a partial order ≤deg on modA(d). There has been a work [1], [10], [11], [21] connecting ≤deg with other partial orders ≤ext and ≤deg on modA(d) defined in terms of extensions and homomorphisms. In particular, it is known that these partial orders coincide in the case A is representation-finite and its Auslander-Reiten quiver is directed. We study degenerations of modules from the additive categories given by connected components of the Auslander-Reiten quiver of A having oriented cycles. We show that the partial orders ≤ext, ≤deg and < coincide on modules from the additive categories of quasi-tubes [24], and describe minimal degenerations of such modules. Moreover, we show that M ≤degN does not imply M ≤ext N for some indecomposable modules M and N lying in coils in the sense of [4].

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