scholarly journals Computations in finite-dimensional Lie algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
A. M. Cohen ◽  
W. A. Graaf ◽  
L. Rónyai
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Collaborative Effort ◽  
Finite Dimensional ◽  
Levi Subalgebra ◽  
International Audience ◽  
Structure Constants ◽  
Technology Foundation ◽  
Finite Dimensional Lie Algebras ◽  
Made In

International audience This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System), within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]). This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.

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

Lie algebras whose maximal subalgebras are modular

1983 ◽  
Vol 94 (1-2) ◽  
pp. 9-13 ◽  
Author(s):  
V. R. Varea
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Maximal Subalgebras ◽  
Modular Element ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

SynopsisA subalgebra M of a Lie algebra L is called modular in L if M is a modular element in the lattice of the subalgebras of L. Our aim is to study the finite-dimensional Lie algebras all of whose maximal subalgebras are modular. We characterize these algebras over any field of characteristic zero.

scholarly journals MULTISTRING VERTICES AND HYPERBOLIC KAC-MOODY ALGEBRAS

1996 ◽  
Vol 11 (03) ◽  
pp. 429-514 ◽  
Author(s):  
R.W. GEBERT ◽  
H. NICOLAI ◽  
P.C. WEST
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Complete Information ◽  
Liouville Theory ◽  
Root Lattice ◽  
Root Space ◽  
Respective Group ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac-Moody algebras, in particular E10. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the N-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds more generally, and may also be valid for uncompactified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain “decoupling polynomials.” This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a nontrivial root space of E10· Because the N-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as E10 by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras.

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.

scholarly journals On normed Lie algebras with sufficiently many subalgebras of codimension I

1986 ◽  
Vol 29 (2) ◽  
pp. 199-220 ◽  
Author(s):  
E. V. Kissin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Arbitrary Field ◽  
Maximal Subalgebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Free Subalgebras ◽  
Image Position ◽  
Infinite Dimensional Lie Algebra ◽  
Finite Dimensional Lie Algebras

Let H be a finite or infinite dimensional Lie algebra. Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras. In [4] complex finite-dimensional Lie algebras were considered for the case when all the maximal subalgebras of H are not necessarily of codimension 1 but whenwhere S(H) is the set of all Lie subalgebras in H of codimension 1. Amayo [1]investigated the finite-dimensional Lie algebras with core-free subalgebras of codimension 1 and also obtained some interesting results about the structure of infinite dimensional Lie algebras with subalgebras of codimension 1.

scholarly journals Lie algebras all of whose maximal subalgebras have codimension one

1981 ◽  
Vol 24 (3) ◽  
pp. 217-219 ◽  
Author(s):  
David Towers
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Arbitrary Field ◽  
Maximal Subalgebras ◽  
Codimension One ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

Let denote the class of finite-dimensional Lie algebras L (over a fixed, but arbitrary, field F) all of whose maximal subalgebras have codimension 1 in L. In (2) Barnes proved that the solvable algebras in are precisely the supersolvable ones. The purpose of this paper is to extend this result and to give a characterisation of all of the algebras in . Throughout we shall place no restrictions on the underlying field of the Lie algebra.

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 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.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document