scholarly journals Graded modules over simple Lie algebras

2019 ◽  
Vol 53 (supl) ◽  
pp. 45-86
Author(s):  
Yuri Bahturin ◽  
Mikhail Kochetov ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Necessary And Sufficient

The paper is devoted to the study of graded-simple modules and gradings on simple modules over finite-dimensional simple Lie algebras. In general, a connection between these two objects is given by the so-called loop construction. We review the main features of this construction as well as necessary and sufficient conditions under which finite-dimensional simple modules can be graded. Over the Lie algebra sl2(C), we consider specific gradings on simple modules of arbitrary dimension.

scholarly journals A remark on irreducible representations of Lie algebras

1979 ◽  
Vol 27 (3) ◽  
pp. 332-336 ◽  
Author(s):  
JU. A. Bahturin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Irreducible Representations ◽  
Bounded Degree ◽  
Precise Form ◽  
Representations Of Lie Algebras ◽  
Necessary And Sufficient

AbstractIn addition to the results of the paper (Bachturin (1974)) we give the precise form of the necessary and sufficient conditions ensuring that all irreducible representations of a Lie algebra were of finite bounded degree.

scholarly journals Post-Lie algebra structures for nilpotent Lie algebras

2018 ◽  
Vol 28 (05) ◽  
pp. 915-933
Author(s):  
Dietrich Burde ◽  
Christof Ender ◽  
Wolfgang Alexander Moens
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Necessary And Sufficient

We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.

On special subalgebras of derivations of Lie algebras

2015 ◽  
Vol 08 (02) ◽  
pp. 1550032 ◽  
Author(s):  
S. Sheikh-Mohseni ◽  
F. Saeedi ◽  
M. Badrkhani Asl
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Let L be a Lie algebra, and Der (L) and IDer (L) be the set of all derivations and inner derivations of L, respectively. Also, let Der c(L) denote the set of all derivations α ∈ Der (L) for which α(x) ∈ Imad x for all x ∈ L. We give necessary and sufficient conditions under which Der c(L) = Der z(L), where Der z(L) is the set of all derivations of L whose images lie in the center of L. Moreover, it is shown that any two isoclinic Lie algebras L1 and L2 satisfy Der c(L1) ≅ Der c(L2).

Graded torsion-free 𝔰𝔩2(ℂ)-modules of rank 2

2021 ◽  
pp. 2250229
Author(s):  
Yuri Bahturin ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Rank 2 ◽  
Torsion Free

In this paper, we explore the possibility of endowing simple infinite-dimensional [Formula: see text]-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in 7, 8.

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

scholarly journals Hom-structures on semi-simple Lie algebras

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Wenjuan Xie ◽  
Quanqin Jin ◽  
Wende Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Simple Lie Algebras ◽  
3 Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Reduction Methods

AbstractA Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

AN W ALGEBRAS AND CLASSIFICATION

1992 ◽  
Vol 07 (36) ◽  
pp. 3419-3423
Author(s):  
LIU CHAO ◽  
BOYU HOU
Keyword(s):  
Lie Algebra ◽  
Regular Element ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arbitrary Degree ◽  
Wznw Model ◽  
Necessary And Sufficient

The necessary and sufficient conditions for the existence of a regular element of arbitrary degree under arbitrary integral gradation of the Lie algebra g is presented. Such elements, while chosen as constraints in WZNW model, give rise to a W-algebra. It is then found that there might be some isomorphic relations between different W-algebras. The necessary conditions for such isomorphisms to appear are also given. Up to the A4 cases these conditions are checked to be sufficient.

Euclidean Lie Algebras

1969 ◽  
Vol 21 ◽  
pp. 1432-1454 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Cartan Matrix ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Generalized Cartan Matrix

Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however.The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.

Titling pair over split-by-nilpotent extensions

2020 ◽  
pp. 2250019
Author(s):  
Dajun Liu ◽  
Jiaqun Wei
Keyword(s):  
Similar Condition ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Split Extension ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Finite Dimensional Algebras

Let [Formula: see text], [Formula: see text] be two finite dimensional algebras over a field [Formula: see text], such that [Formula: see text] is a split extension of A by the nilpotent bimodule [Formula: see text]. We mainly give necessary and sufficient conditions for a tilting pair [Formula: see text] such that [Formula: see text] or [Formula: see text] are tilting pairs. Also, we obtain a similar condition such that a Wakamatsu tilting pair [Formula: see text] in [Formula: see text]-mod can be a Wakamatsu tilting pair [Formula: see text] in [Formula: see text]-mod.

On Derivations of Lie Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 174-180 ◽  
Author(s):  
Stephen Berman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Cartan Matrices

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

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