scholarly journals ON A FORMULA FOR THE PI-EXPONENT OF LIE ALGEBRAS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350069 ◽  
Author(s):  
A. S. GORDIENKO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Formula ◽  
Natural Generalization ◽  
Rational Action ◽  
Semisimple Lie Algebra ◽  
Arbitrary Group ◽  
Solvable Radical ◽  
Finite Dimensional ◽  
Algebra Over A Field

We prove that one of the conditions in Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite-dimensional semisimple Lie algebra acts by derivations on a finite-dimensional Lie algebra over a field of characteristic 0, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial G-identities of a finite-dimensional Lie algebra with a rational action of a connected reductive affine algebraic group G by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for H-module Lie algebras whose solvable radical is nilpotent, assuming only the H-invariance of the radical, i.e. under weaker assumptions on the H-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for G-codimensions of all finite-dimensional Lie G-algebras whose solvable radical is nilpotent, for an arbitrary group G.

scholarly journals Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

2015 ◽  
Vol 15 (02) ◽  
pp. 1650029 ◽  
Author(s):  
Leandro Cagliero ◽  
Fernando Szechtman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Weight Space ◽  
Arbitrary Field ◽  
Nilpotent Radical ◽  
Indecomposable Modules ◽  
Finite Dimensional ◽  
A Chain ◽  
Algebra Over A Field

Let 𝔤 be a finite-dimensional Lie algebra over a field of characteristic 0, with solvable radical 𝔯 and nilpotent radical 𝔫 = [𝔤, 𝔯]. Given a finite-dimensional 𝔤-module U, its nilpotency series 0 ⊂ U(1) ⊂ ⋯ ⊂ U(m) = U is defined so that U(1) is the 0-weight space of 𝔫 in U, U(2)/U(1) is the 0-weight space of 𝔫 in U/U(1), and so on. We say that U is linked if each factor of its nilpotency series is a uniserial 𝔤/𝔫-module, i.e. its 𝔤/𝔫-submodules form a chain. Every uniserial 𝔤-module is linked, every linked 𝔤-module is indecomposable with irreducible socle, and both converses fail. In this paper, we classify all linked 𝔤-modules when 𝔤 = 〈x〉 ⋉ 𝔞 and ad x acts diagonalizably on the abelian Lie algebra 𝔞. Moreover, we identify and classify all uniserial 𝔤-modules amongst them.

scholarly journals Some remarks on graded nilpotent Lie algebras and the Toral Rank Conjecture

2014 ◽  
Vol 14 (02) ◽  
pp. 1550024
Author(s):  
Guillermo Ames ◽  
Leandro Cagliero ◽  
Mónica Cruz
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Algebraic Version ◽  
Finite Dimensional ◽  
Rank Conjecture ◽  
Algebra Over A Field ◽  
Toral Rank

If 𝔫 is a [Formula: see text]-graded nilpotent finite-dimensional Lie algebra over a field of characteristic zero, a well-known result of Deninger and Singhof states that dim H*(𝔫) ≥ L(p) where p is the polynomial associated to the grading and L(p) is the sum of the absolute values of the coefficients of p. From this result they derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that dim H*(𝔫) ≥ 2 dim (ℨ) for any finite-dimensional nilpotent Lie algebra 𝔫 with center ℨ. The TRC is more than 25 years old and remains open even for [Formula: see text]-graded 3-step nilpotent Lie algebras. Investigating to what extent the bound given by Deninger and Singhof could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra 𝔫 with center ℨ: (A) If 𝔫 admits a [Formula: see text]-grading [Formula: see text], such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a ℤ+-grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? (B) If 𝔫 is r-step nilpotent admitting a grading 𝔫 = 𝔫1 ⊕ 𝔫2 ⊕ ⋯ ⊕ 𝔫k such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.

A NEW CLASS OF INFINITE-DIMENSIONAL LIE ALGEBRAS: AN ANALYTICAL CONTINUATION OF THE ARBITRARY FINITE-DIMENSIONAL SEMISIMPLE LIE ALGEBRA

1990 ◽  
Vol 05 (15) ◽  
pp. 1167-1174 ◽  
Author(s):  
E. S. FRADKIN ◽  
V. YA. LINETSKY
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Root Lattice ◽  
Semisimple Lie Algebra ◽  
Infinite Dimensional ◽  
New Class ◽  
Finite Dimensional ◽  
Higher Spin ◽  
Structure Constants ◽  
Infinite Dimensional Lie Algebra

With any semisimple Lie algebra g we can associate an infinite-dimensional Lie algebra AC (g) which is an analytic continuation of g from its root system to its root lattice. The manifest expressions for the structure constants of analytic continuations of the symplectic Lie algebras sp 2n are obtained by the Poisson-bracket realizations method and AC (g) for g = sl n and so n are discussed. The representations, central extension, supersymmetric and higher spin generalizations are considered. The Virasoro theory is a particular case where g = sp 2.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Vector bundles associated to Lie algebras

2016 ◽  
Vol 2016 (716) ◽  
Author(s):  
Jon F. Carlson ◽  
Eric M. Friedlander ◽  
Julia Pevtsova
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Bundles ◽  
Coherent Sheaves ◽  
Restricted Lie Algebra ◽  
Finite Dimensional

AbstractWe introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra

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

scholarly journals Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex

2019 ◽  
Vol 18 (12) ◽  
pp. 1950233 ◽  
Author(s):  
Farrukh Mukhamedov ◽  
Otabek Khakimov ◽  
Bakhrom Omirov ◽  
Izzat Qaralleh
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimensional ◽  
Evolution Algebra

This paper is devoted to the nilpotent finite-dimensional evolution algebras [Formula: see text] with [Formula: see text]. We describe the Lie algebra of derivations of these algebras. Moreover, in terms of these Lie algebras, we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. Besides, all automorphisms and local automorphisms of these algebras are found.

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 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.

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