scholarly journals The Wielandt Subalgebra of a Lie Algebra

2003 ◽  
Vol 74 (3) ◽  
pp. 313-330 ◽  
Author(s):  
Donald W. Barnes ◽  
Daniel Groves
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Ground Field ◽  
Derived Length ◽  
Finite Dimensional ◽  
Definition Of

AbstractFollowing the analogy with group theory, we define the Wielandt subalgebra of a finite-dimensional Lie algebra to be the intersection of the normalisers of the subnormal subalgebras. In a non-zero algebra, this is a non-zero ideal if the ground field has characteristic 0 or if the derived algebra is nilpotent, allowing the definition of the Wielandt series. For a Lie algebra with nilpotent derived algebra, we obtain a bound for the derived length in terms of the Wielandt length and show this bound to be best possible. We also characterise the Lie algebras with nilpotent derived algebra and Wielandt length 2.

On Covers of Perfect Lie Algebras

2011 ◽  
Vol 18 (03) ◽  
pp. 419-427 ◽  
Author(s):  
Ali Reza Salemkar ◽  
Behrouz Edalatzadeh ◽  
Hamid Mohammadzadeh
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Central Extension ◽  
Complete Structure ◽  
Finite Dimensional ◽  
Equivalent Conditions

A Lie algebra is said to be perfect when it coincides with its derived subalgebra. The paper is devoted to give a complete structure of covers of perfect Lie algebras. Also, similar to a result of Alperin and Gorenstein (1966) in group theory, it is shown that every automorphism of a finite dimensional perfect Lie algebra may be lifted to an automorphism of its cover. Moreover, we present the concepts of irreducible and primitive extensions of an arbitrary Lie algebra and give some equivalent conditions for a central extension to be irreducible or primitive. Finally, we study the connection between the primitive extensions and the stem covers of finite dimensional perfect Lie algebras.

scholarly journals PLAIN REPRESENTATIONS OF LIE ALGEBRAS

2001 ◽  
Vol 63 (3) ◽  
pp. 571-591 ◽  
Author(s):  
A. A. BARANOV ◽  
A. E. ZALESSKII
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
General Problem ◽  
Associative Algebras ◽  
Ground Field ◽  
Finite Dimensional ◽  
Representations Of Lie Algebras ◽  
Completely Reducible ◽  
Finite Dimensional Lie Algebras

In this paper we study representations of finite dimensional Lie algebras. In this case representations are not necessarily completely reducible. As the general problem is known to be of enormous complexity, we restrict ourselves to representations that behave particularly well on Levi subalgebras. We call such representations plain (Definition 1.1). Informally, we show that the theory of plain representations of a given Lie algebra L is equivalent to representation theory of finitely many finite dimensional associative algebras, also non-semisimple. The sense of this is to distinguish representations of Lie algebras that are of complexity comparable with that of representations of associative algebras. Non-plain representations are intrinsically much more complex than plain ones. We view our work as a step toward understanding this complexity phenomenon.We restrict ourselves also to perfect Lie algebras L, that is, such that L = [L, L]. In our main results we assume that L is perfect and [sfr ][lfr ]2-free (which means that L has no quotient isomorphic to [sfr ][lfr ]2). The ground field [ ] is always assumed to be algebraically closed and of characteristic 0.

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 Weyl modules and Weyl functors for hyper-map algebras

2020 ◽  
pp. 2140007
Author(s):  
Angelo Bianchi ◽  
Samuel Chamberlin
Keyword(s):  
Lie Algebra ◽  
Finitely Generated ◽  
Weyl Modules ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Universal Properties ◽  
Definition Of ◽  
Unitary Algebra

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

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 Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1737
Author(s):  
Mariia Myronova ◽  
Jiří Patera ◽  
Marzena Szajewska
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Coxeter Groups ◽  
Reflection Groups ◽  
Geometrical Structures ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Branching Rules ◽  
Definition Of ◽  
Representation Orbit

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

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.

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