scholarly journals Representations of Bihom-Lie Algebras

2022 ◽  
Vol 29 (01) ◽  
pp. 125-142
Author(s):  
Yongsheng Cheng ◽  
Huange Qi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Adjoint Representation ◽  
Trivial Representation ◽  
Central Extensions ◽  
Linear Maps

A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.

scholarly journals About Zero Torsion Linear Maps on Lie Algebras

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Louis Magnin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Equivalence Classes ◽  
Explicit Description ◽  
Complex Structures ◽  
Linear Map ◽  
3 Dimensional ◽  
Linear Maps

We prove that any zero torsion linear map on a nonsolvable real Lie algebra is an extension of some CR-structure. We then study the cases of (2, ) and the 3-dimensional Heisenberg Lie algebra . In both cases, we compute up to equivalence all zero torsion linear maps on , and deduce an explicit description of the equivalence classes of integrable complex structures on .

Representations and cohomologies of regular Hom-pre-Lie algebras

2019 ◽  
Vol 19 (08) ◽  
pp. 2050149
Author(s):  
Shanshan Liu ◽  
Lina Song ◽  
Rong Tang
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Theory ◽  
Trivial Representation ◽  
Linear Deformation ◽  
Dual Representations ◽  
Regular Representations ◽  
Equivalent Characterization ◽  
Tensor Representations

In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of regular Hom-pre-Lie algebras in terms of the cohomology theory of regular Hom-Lie algebras. As applications, we study linear deformations of regular Hom-pre-Lie algebras, which are characterized by the second cohomology groups of regular Hom-pre-Lie algebras with the coefficients in the regular representations. The notion of a Nijenhuis operator on a regular Hom-pre-Lie algebra is introduced which can generate a trivial linear deformation of a regular Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a regular Hom-pre-Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an [Formula: see text]-operator on a regular Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure.

MATHEMATICA™ PACKAGES FOR COMPUTING PRINCIPAL DECOMPOSITIONS OF SIMPLE LIE ALGEBRAS AND APPLICATIONS IN EXTENDED CONFORMAL FIELD THEORIES

2003 ◽  
Vol 14 (01) ◽  
pp. 1-27 ◽  
Author(s):  
DANIELA GĂRĂJEU ◽  
MIHAIL GĂRĂJEU
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Adjoint Representation ◽  
Cartan Matrix ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Conformal Field

In this article, we propose two Mathematica™ packages for doing calculations in the domain of classical simple Lie algebras. The main goal of the first package, [Formula: see text], is to determine the principal three-dimensional subalgebra of a simple Lie algebra. The package provides several functions which give some elements related to simple Lie algebras (generators in fundamental and adjoint representation, roots, Killing form, Cartan matrix, etc.). The second package, [Formula: see text], concerns the principal decomposition of a Lie algebra with respect to the principal three-dimensional embedding. These packages have important applications in extended two-dimensional conformal field theories. As an example, we present an application in the context of the theory of W-gravity.

scholarly journals Invariant metrics on central extensions of quadratic Lie algebras

2019 ◽  
Vol 19 (12) ◽  
pp. 2050224
Author(s):  
R. García-Delgado ◽  
G. Salgado ◽  
O. A. Sánchez-Valenzuela
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
General Structure ◽  
Invariant Metrics ◽  
Invariant Metric ◽  
Central Extensions ◽  
Direct Sum ◽  
Quadratic Lie Algebra

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.

Some Properties of the c-Nilpotent Multiplier and c-Covers of Lie Algebras

2014 ◽  
Vol 21 (03) ◽  
pp. 421-426 ◽  
Author(s):  
Mohammad Reza Rismanchian ◽  
Mehdi Araskhan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extensions ◽  
Finite Dimensional

This paper is devoted to present some properties of the c-nilpotent multiplier [Formula: see text] and some features of c-central extensions of a (finite dimensional) Lie algebra L. Moreover, we give the structure of all c-covers of Lie algebras whose c-nilpotent multipliers have the Hopfian property.

Dual Space Derivations and H2(L, F) of Modular Lie Algebras

1987 ◽  
Vol 39 (5) ◽  
pp. 1078-1106 ◽  
Author(s):  
Rolf Farnsteiner
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Dual Space ◽  
Positive Characteristic ◽  
Base Field ◽  
Central Extensions ◽  
Simple Lie Algebras ◽  
Early Results ◽  
Modular Lie Algebras

It is well-known that the classical vanishing results of the cohomology theory of Lie algebras depend on the characteristic of the underlying base field. The theorems of Cartan and Zassenhaus, for instance, entail that non-modular simple Lie algebras do not admit non-trivial central extensions. In contrast, early results by Block [3] prove that this conclusion loses its validity if the underlying base field has positive characteristic.Central extensions of a given Lie algebra L, or equivalently its second cohomology group H(L, F), can be conveniently described by means of derivations φ:L → L*.

scholarly journals Structure of the Extended Schrödinger–Virasoro Lie Algebra $\widetilde{\mathfrak{sv}}$

2009 ◽  
Vol 16 (04) ◽  
pp. 549-566 ◽  
Author(s):  
Shoulan Gao ◽  
Cuipo Jiang ◽  
Yufeng Pei
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Statistical Physics ◽  
Two Dimensional ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Infinite Dimensional ◽  
Conformal Field

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra [Formula: see text], introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that [Formula: see text] is an infinite-dimensional complete Lie algebra, and the universal central extension of [Formula: see text] in the category of Leibniz algebras is the same as that in the category of Lie algebras.

INFINITE-DIMENSIONAL ℤN-INDEXED LIE ALGEBRAS AND THEIR SUPERSYMMETRIC GENERALIZATIONS

1989 ◽  
Vol 04 (16) ◽  
pp. 4295-4302 ◽  
Author(s):  
EDUARDO RAMOS ◽  
ROBERT E. SHROCK
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Explicit Representation ◽  
Mathematical Basis ◽  
Central Extensions ◽  
Infinite Dimensional ◽  
Different Types ◽  
Structure Constants ◽  
Infinite Dimensional Lie Algebra

We exhibit a new infinite-dimensional Lie algebra involving operators Ln with indices n ∈ ℤN and depending on a vector of structure constants, v. Two different types of central extensions are also presented. By constructing an explicit representation, we show that this algebra has a natural mathematical basis in terms of the algebra of infinitesimal diffeomorphisms of (S1)N. For N ≥ 2, the space of states is shown to have properties very different from those of the N = 1 (Virasoro) case. Supersymmetric generalizations are also given.

scholarly journals Topics on n-ary Algebraic Structures

10.14311/1179 ◽  
2010 ◽  
Vol 50 (3) ◽  
Author(s):  
J. A. de Azcárraga ◽  
J. M. Izquierdo
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Structures ◽  
Central Extensions ◽  
Generalized Poisson ◽  
Leibniz Algebras ◽  
Infinitesimal Deformations ◽  
Generalized Lie Algebras

We review the basic definitions and properties of two types of n-ary structures, the Generalized Lie Algebras (GLA) and the Filippov (≡ n-Lie) algebras (FA), as well as those of their Poisson counterparts, the Generalized Poisson (GPS) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology complexes relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends Whitehead’s lemma to all n ≥ 2, n = 2 being the original Lie algebra case. Some comments onn-Leibniz algebras are also made.

scholarly journals Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

2020 ◽  
Author(s):  
Jun Jiang ◽  
◽  
Satyendra Kumar Mishra ◽  
Yunhe Sheng ◽  
◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Group Action ◽  
Lie Group ◽  
Smooth Manifold ◽  
Adjoint Representation ◽  
Lie Group Action

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra (gl(V),[.,.],Ad), and the derivation Hom-Lie algebra of a Hom-Lie algebra.

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