Lie Algebras with Nilpotent Centralizers

1979 ◽  
Vol 31 (5) ◽  
pp. 929-941 ◽  
Author(s):  
G. M. Benkart ◽  
I. M. Isaacs
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebras ◽  
Arbitrary Characteristic ◽  
Algebra For All ◽  
Finite Dimensional ◽  
Nilpotent Subalgebra ◽  
Finite Dimensional Lie Algebras

We consider finite dimensional Lie algebras over an algebraically closed field F of arbitrary characteristic. Such an algebra L will be called a centralizer nilpotent Lie algebra (abbreviated c.n.) provided that the centralizer C(x) is a nilpotent subalgebra of L for all nonzero x ∈ L.For each algebraically closed F, there is a unique simple Lie algebra of dimension 3 over F which we shall denote S(F). This algebra has a basis e−1, e0, e1 such that [e−1e0] = e−1, [e−1e1] = e0 and [e0e1] = e1. (If char(F) ≠ 2, then S(F) ≅ sl2(F).) It is trivial to check that S(F) is a c.n. algebra for all F.There are two other types of simple Lie algebras we consider. If char (F) = 3, construct the octonion (Cayley) algebra over F.

scholarly journals Lie Algebras with Abelian Centralizers

2010 ◽  
Vol 17 (04) ◽  
pp. 629-636 ◽  
Author(s):  
Igor Klep ◽  
Primož Moravec
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Closed Field ◽  
Split Extension ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Solvable Case ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

We classify all finite-dimensional Lie algebras over an algebraically closed field of characteristic 0, whose nonzero elements have abelian centralizers. These algebras are either simple or solvable, where the only simple such Lie algebra is [Formula: see text]. In the solvable case, they are either abelian or a one-dimensional split extension of an abelian Lie algebra.

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.

scholarly journals Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

2008 ◽  
Vol 11 ◽  
pp. 280-297 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Algorithm ◽  
Adjoint Action ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Simple Lie Algebras

AbstractLet G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.

scholarly journals On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

1965 ◽  
Vol 25 ◽  
pp. 211-220 ◽  
Author(s):  
Hiroshi Kimura
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional

Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known; 1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,2) Determination of all irreducible representations in connection with their highest weights.3) Weyl’s formula for the character of irreducible representations [9].4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],5) The law of the decomposition of the tensor product of two irreducible representations [1].

scholarly journals Group Gradings on Finite Dimensional Lie Algebras

2013 ◽  
Vol 20 (04) ◽  
pp. 573-578 ◽  
Author(s):  
Dušan Pagon ◽  
Dušan Repovš ◽  
Mikhail Zaicev
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Solvable Radical ◽  
Finite Dimensional ◽  
Levi Subalgebra ◽  
Group Gradings ◽  
Finite Dimensional Lie Algebras

We study gradings by non-commutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if L is graded by a non-abelian finite group G, then the solvable radical R of L is G-graded and there exists a Levi subalgebra B=H1⊕ ⋯ ⊕ Hm homogeneous in G-grading with graded simple summands H1,…,Hm. All Supp Hi (i=1,…,m) are commutative subsets of G.

Codimension growth of central polynomials of Lie algebras

2020 ◽  
Vol 32 (1) ◽  
pp. 201-206
Author(s):  
Antonio Giambruno ◽  
Mikhail Zaicev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Adjoint Representation ◽  
Closed Field ◽  
Polynomial Identities ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Linearly Independent

AbstractLet L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and let I be the T-ideal of polynomial identities of the adjoint representation of L. We prove that the number of multilinear central polynomials in n variables, linearly independent modulo I, grows exponentially like {(\dim L)^{n}}.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

scholarly journals k-step nilpotent Lie algebras

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Michel Goze ◽  
Elisabeth Remm
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Deformation Process ◽  
Nilpotent Lie Algebras ◽  
Early Stages ◽  
Finite Dimensional ◽  
Contraction Process ◽  
Finite Dimensional Lie Algebras

AbstractThe classification of complex or real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example, the nilpotent Lie algebras are classified only up to dimension 7. Moreover, to recognize a given Lie algebra in the classification list is not so easy. In this work, we propose a different approach to this problem. We determine families for some fixed invariants and the classification follows by a deformation process or a contraction process. We focus on the case of 2- and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology for this type of algebras and the algebras which are rigid with respect to this cohomology. Other

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

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