Pilar Benito - Academia.edu (original) (raw)

Papers by Pilar Benito

Bulletin of the Brazilian Mathematical Society, New Series

The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with c... more The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with chain lattices of ideals are elementary blocks in the embedding or decomposition of Lie algebras with finite lattice of ideals. Secondly, to show that the number of Lie algebras of this class is large and they support other types of Lie structures. Beginning with general examples and algebraic decompositions, we focus on computational algorithms to build Lie algebras in which the lattice of ideals is a chain. The chain condition forces gradings on the nilradicals of this class of algebras. Our algorithms yield to several positive naturally graded parametric families of Lie algebras. Further generalizations and other kind of structures will also be discussed.

Linear Algebra and its Applications

The double extension and the T *-extension are classical methods for constructing finite dimensio... more The double extension and the T *-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.

arXiv: Rings and Algebras, 2019

Let nd,t\n_{d,t}nd,t be the free nilpotent Lie algebra of type ddd and nilindex ttt. Starting out with ... more Let nd,t\n_{d,t}nd,t be the free nilpotent Lie algebra of type ddd and nilindex ttt. Starting out with the derivation algebra and the automorphism group of nd,t\n_{d,t}nd,t, we get a natural description of derivations and automorphisms of any generic nilpotent Lie algebra of the same type and nilindex. Moreover, along the paper we discuss several examples to illustrate the obtained results.

Any nilpotent Lie algebra is a quotient of a free nilpotent Lie algebra of the same nilindex and ... more Any nilpotent Lie algebra is a quotient of a free nilpotent Lie algebra of the same nilindex and type. In this paper we review some nice features of the class of free nilpotent Lie algebras. We will focus on the survey of Lie algebras of derivations and groups of automorphisms of this class of algebras. Three research projects on nilpotent Lie algebras will be mentioned.

Nonassociative algebra and its applications, 2019

Mediterranean Journal of Mathematics, 2018

We prove that all Rota-Baxter operators on a quadratic division algebra are trivial. For nonzero ... more We prove that all Rota-Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota-Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota-Baxter operators and the solutions to the alternative Yang-Baxter equation on the Cayley-Dickson algebra. We also investigate the Rota-Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.

$Research paper thumbnail of Sage computations of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi mathvariant="fraktur">s</mi><mi mathvariant="fraktur">l</mi></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{sl}_2(k)</annotation></semantics></math>sl2(k)-Levi extensions$

Tbilisi Mathematical Journal, 2012

In 2010, Snolb [9] studied the structure of nilpotent Lie algebras admitting a Levi extension. As... more In 2010, Snolb [9] studied the structure of nilpotent Lie algebras admitting a Levi extension. As a corollary of the results therein, it is shown that the classes of characteristically nilpotent or filiform Lie algebras do not admit Levi extensions. The paper ends by asking for the possibility of finding series of nilpotent Lie algebras in arbitrary dimension not being abelian or Heisenberg and allowing such extensions. Our goal in this work is to present computational examples of this type of algebras by using Sage software. In the case of nilpotent Lie algebras admitting sl2(k) as Levi factor special constructions will be given by means of Sage routines based on transvections over sl2(k)-irreducible modules.

Journal of Symbolic Computation, 2019

Taking into account the theoretical results and guidelines given in this work, we introduce a com... more Taking into account the theoretical results and guidelines given in this work, we introduce a computational method to construct any 2step nilpotent quadratic algebra of d generators. Along the work we show that the key of the classification of this class of metric algebras relies on certain families of skewsymmetric matrices. Computational examples for d ≤ 8 will be given.

Linear Algebra and its Applications, 2017

In this paper we introduce an equivalence between the category of the tnilpotent quadratic Lie al... more In this paper we introduce an equivalence between the category of the tnilpotent quadratic Lie algebras with d generators and the category of some symmetric invariant bilinear forms over the t-nilpotent free Lie algebra with d generators. Taking into account this equivalence, t-nilpotent quadratic Lie algebras with d generators are classified (up to isometric isomorphisms, and over any field of characteristic zero), in the following cases: d = 2 and t ≤ 5, d = 3 and t ≤ 3.

A large number of algebraic structures, among which the associative and the Jordan algebras deser... more A large number of algebraic structures, among which the associative and the Jordan algebras deserve special mention, are closely related to the Lie algebras and to some interesting geometries. These relationships explain certain exceptional behaviors in Algebra and Geometry, which are nothing but manifestations of the same phenomena. In this note we analyze part of the research made during the last years by the research group of Algebra of the University of La Rioja. This research focuses on the study of non associative structures related to Lie algebras.

We propose the study and description of the structure of complex Lie algebras with nilradical a n... more We propose the study and description of the structure of complex Lie algebras with nilradical a nilpotent Lie algebra of type 2 by using sl2(C)-representation theory. Our results will be applied to review the classi?cation given in [1] (J. Geometry and Physics, 2011) of the Lie algebras with nilradical the quasiclassical algebra L5;3. A non-Lie algebra has been erroneously included in this classi?cation. The 5-dimensional Lie algebra L5;3 is a free nilpotent algebra of type 2 and it is one of two free nilpotent algebras admitting an invariant metric. According to [, Ok98] quasiclassical algebras let construct consistent Yang-Mills gauge theories.

Contemporary Mathematics, 2009

Contemporary Mathematics Volume 483, 2009 Tits construction, triple systems and pairs Pilar Benit... more Contemporary Mathematics Volume 483, 2009 Tits construction, triple systems and pairs Pilar Benito and Fabian Martın-Herce Abstract. The paper is concerned with the description of certain well-known models of triple systems and simple Jordan pairs by reviewing a slightly gen- ...

Linear and Multilinear Algebra, 2014

Linear Algebra and its Applications, 2003

Different models of the Cayley algebras and of their Lie algebras of derivations are given, based... more Different models of the Cayley algebras and of their Lie algebras of derivations are given, based on some distinguished subalgebras of the later ones.

Linear Algebra and its Applications, 2013

Levi's theorem decomposes any arbitrary Lie algebra over a field of characteristic zero, as a dir... more Levi's theorem decomposes any arbitrary Lie algebra over a field of characteristic zero, as a direct sum of a semisimple Lie algebra (named Levi factor) and its solvable radical. Given a solvable Lie algebra R, a semisimple Lie algebra S is said to be a Levi extension of R in case a Lie structure can be defined on the vector space S ⊕ R. The assertion is equivalent to ρ(S) ⊆ Der(R), where Der(R) is the derivation algebra of R, for some representation ρ of S onto R. Our goal in this paper, is to present some general structure results on nilpotent Lie algebras admitting Levi extensions based on free nilpotent Lie algebras and modules of semisimple Lie algebras. In low nilpotent index a complete classification will be given. The results are based on linear algebra methods and leads to computational algorithms

Journal of Pure and Applied Algebra, 2009

Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately ... more Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their Lie inner derivation algebra are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems will be shown to split into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types will be classified and most of them will be shown to be related to a Generalized Tits Construction of Lie algebras.

Journal of Pure and Applied Algebra, 2005

Lie-Yamaguti algebras (or generalized Lie triple systems) are intimately related to reductive hom... more Lie-Yamaguti algebras (or generalized Lie triple systems) are intimately related to reductive homogeneous spaces. Simple Lie-Yamaguti algebras whose standard enveloping Lie algebra is the simple Lie algebra of type G2 are described, making use of the octonions. These examples reveal the much greater complexity of these systems, compared to Lie triple systems.

Journal of Algebra, 2012

In 2002, T.L. Hodge and B.J. Parshall [7] overviewed the representation theory of Lie triple syst... more In 2002, T.L. Hodge and B.J. Parshall [7] overviewed the representation theory of Lie triple systems (Lts for short). They proved that finite-dimensional modules of Lts in the sense of Harris (1961) [5] can be described by using involutory modules of their universal enveloping Lie algebra. The main goal of this paper is to explore the dimension of irreducible modules for simple Lts through dimensional formulas based on the remarkable Weyl formula of irreducible modules of simple Lie algebras. The paper also includes the complete classification of one-dimensional modules in arbitrary characteristic. These modules are the infinitesimal analog of symmetric line bundles.

Journal of Algebra, 1999

Lie triple system T over a field F of characteristic zero. It turns out that it contains nontrivi... more Lie triple system T over a field F of characteristic zero. It turns out that it contains nontrivial elements if and only if T is related to a simple Jordan algebra. In particular this provides a new proof of the determination by Laquer of the invariant affine connections in the simply connected compact irreducible Riemannian symmetric spaces.

Communications in Algebra, 2010

We prove that there are no simple commutative n-ary Leibniz algebras of arbitrary dimension over ... more We prove that there are no simple commutative n-ary Leibniz algebras of arbitrary dimension over fields of characteristic zero or greater than n. This result extends previous work of A.P. Pojidaev and A. Elduque.