Hom-structures on simple graded Lie algebras of finite growth

2016 ◽  
Vol 16 (08) ◽  
pp. 1750154 ◽  
Author(s):  
Wenjuan Xie ◽  
Wende Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jordan Algebra ◽  
Jacobi Identity ◽  
Direct Consequence ◽  
Classification Theorem ◽  
Algebra Homomorphism ◽  
Graded Lie Algebras ◽  
Linear Map ◽  
Finite Growth

A Hom-structure on a Lie algebra [Formula: see text] is a linear map [Formula: see text] satisfying the Hom–Jacobi identity: [Formula: see text] for all [Formula: see text]. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. In this paper, using a classification theorem due to Mathieu, we determine explicitly all the Hom-structures on the simple graded Lie algebras of finite growth. As a direct consequence, all the Hom-structures on any simple graded Lie algebras of finite growth constitute a Jordan algebra in the usual way.

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.

Generators of simple graded Lie algebras of finite growth

2017 ◽  
Vol 148 (2) ◽  
pp. 315-324
Author(s):  
Liming Tang ◽  
Wende Liu
Keyword(s):  
Lie Algebras ◽  
Graded Lie Algebras ◽  
Finite Growth

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 .

ON THE STRUCTURE OF SOLUBLE GRADED LIE ALGEBRAS

2011 ◽  
Vol 10 (04) ◽  
pp. 597-604 ◽  
Author(s):  
PAVEL SHUMYATSKY ◽  
CARMELA SICA
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Graded Lie Algebras ◽  
Derived Length ◽  
Elementary Group ◽  
Graded Lie Algebra

Let A be the elementary group of order 2n and L an A-graded Lie algebra with L0 = 0. Assume that L is soluble with derived length k. It is proved that L has a series of ideals of length n all of whose quotients are nilpotent of {k, n}-bounded class.

q-Deformations of 3-Lie Algebras

2017 ◽  
Vol 24 (03) ◽  
pp. 519-540 ◽  
Author(s):  
Ruipu Bai ◽  
Lixin Lin ◽  
Yan Zhang ◽  
Chuangchuang Kang
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extension ◽  
Jacobi Identity ◽  
Group Algebras ◽  
Type I ◽  
Associative Algebras

q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra [Formula: see text], where [Formula: see text] and [Formula: see text], is a vector space A over a field 𝔽 with 3-ary linear multiplications [ , , ]q and [Formula: see text] from [Formula: see text] to A, and a map [Formula: see text] satisfying the q-Jacobi identity [Formula: see text] for all [Formula: see text]. If the multiplications satisfy that [Formula: see text] and [Formula: see text] is skew-symmetry, then [Formula: see text] is called a type (I)-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type (I)-q-3-Lie algebras are constructed. Also, the non-trivial onedimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras [Formula: see text], and [Formula: see text] are obtained.

scholarly journals On Some Structural Components of Nilsolitons

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Hulya Kadioglu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Sufficient Conditions ◽  
Structural Components ◽  
Nilpotent Lie Algebras ◽  
Structure Constants ◽  
Nilsoliton Metric

In this paper, we study nilpotent Lie algebras that admit nilsoliton metric with simple pre-Einstein derivation. Given a Lie algebra η , we would like to compute as much of its structure as possible. The structural components we consider in this study are the structure constants, the index, and the rank of the nilsoliton derivations. For this purpose, we prove necessary or sufficient conditions for an algebra to admit such metrics. Particularly, we prove theorems for the computation of the Jacobi identity for a given algebra so that we can solve the system of the equation(s) and find the structure constants of the nilsoliton.

scholarly journals Lie and Jordan structure in operator algebras

1980 ◽  
Vol 29 (2) ◽  
pp. 129-142
Author(s):  
Derek W. Robinson ◽  
Erling Størmer
Keyword(s):  
Hilbert Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Jordan Algebra ◽  
Operator Algebras ◽  
Bounded Operators ◽  
Jordan Structure ◽  
Spectral Subspaces

AbstractLet υ be a C*-algebra, α a *-anti-automorphism of order 2, and υα(±1) = {A; A ∈ υ, α(A) = ± A} the spectral subspaces of α. It follows that υα(+ 1) is a Jordan algebra and υα(− 1) is a Lie algebra. We begin the classification of pairs of Jordan and Lie algebras which can occur in this manner by examining υ = ℒ(ℋ), the algebra of bounded operators on a Hilbert space ℋ.

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