scholarly journals On involutive Lie algebras having a Cartan decomposition

2004 ◽  
Vol 69 (2) ◽  
pp. 191-202 ◽  
Author(s):  
A. J. Calderón Martín
Keyword(s):  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Direct Limit ◽  
Arbitrary Dimension ◽  
Cartan Decomposition ◽  
Infinite Dimensional ◽  
Finite Dimensional

We introduce the concept of Cartan decomposition relative to a Cartan subalgebra H in the sense of Y. Billig and A. Pianzola for involutive complex Lie algebras L of arbitrary dimension. If L has such a decomposition and is infinite dimensional and simple, we show it is *-isomorphic to a direct limit of classical finite dimensional simple involutive Lie algebras of the same type A, B, C, or D.

scholarly journals Hereditary automorphic Lie algebras

2019 ◽  
Vol 22 (08) ◽  
pp. 1950076
Author(s):  
Vincent Knibbeler ◽  
Sara Lombardo ◽  
Jan A. Sanders
Keyword(s):  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Dimensional Space ◽  
Projective Line ◽  
Killing Form ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Novel Approach ◽  
Automorphic Functions

We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line.

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.

scholarly journals Soluble Lie algebras having finite-dimensional maximal subalgebras

1974 ◽  
Vol 11 (1) ◽  
pp. 145-156 ◽  
Author(s):  
Ian N. Stewart
Keyword(s):  
Lie Algebras ◽  
Minimal Ideal ◽  
Prime Characteristic ◽  
Maximal Subalgebra ◽  
Maximal Subalgebras ◽  
Infinite Dimensional ◽  
The Core ◽  
Zero Characteristic ◽  
Finite Dimensional ◽  
Split Extensions

Infinite-dimensional soluble Lie algebras can possess maximal subalgebras which are finite-dimensional. We give a fairly complete description of such algebras: over a field of prime characteristic they do not exist; over a field of zero characteristic then, modulo the core of the aforesaid maximal subalgebra, they are split extensions of an abelian minimal ideal by the maximal subalgebra. If the field is algebraically closed, or if the maximal subalgebra is supersoluble, then all finite-dimensional maximal subalgebras are conjugate under the group of automorphisms generated by exponentials of inner derivations by elements of the Fitting radical. An example is given to indicate the differences encountered in the insoluble case, and the nonexistence of group-theoretic analogues is briefly discussed.

DEFORMATIONS OF LOOP ALGEBRAS AND CLASSICAL INTEGRABLE SYSTEMS: FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

2004 ◽  
Vol 16 (07) ◽  
pp. 823-849 ◽  
Author(s):  
T. SKRYPNYK
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Finite Dimensional ◽  
Quasigraded Lie Algebras ◽  
Classical Integrable

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

scholarly journals A Comparison of Deformations and Geometric Study of Varieties of Associative Algebras

2007 ◽  
Vol 2007 ◽  
pp. 1-24 ◽  
Author(s):  
Abdenacer Makhlouf
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Noncommutative Version ◽  
The Common ◽  
Global Deformation ◽  
Formal Power ◽  
Deformation Approach ◽  
Geometric Study

The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. In each case we describe the corresponding notions of degeneration and rigidity. We illustrate these notions by examples and give some general properties. The last part of this work shows how these notions help in the study of varieties of associative algebras. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general approach called global deformation follows from a general theory by Schlessinger and was developed by A. Fialowski in order to deform infinite-dimensional nilpotent Lie algebras. In a nonstandard framework, M. Goze introduced the notion of perturbation for studying the rigidity of finite-dimensional complex Lie algebras. All these approaches share the common fact that we make an “extension” of the field. These theories may be applied to any multilinear structure. In this paper, we will be dealing with the category of associative algebras.

scholarly journals K-loop Structures Raised by the Direct Limits of Pseudo Unitary $U(p, a_{n})$ and Pseudo Orthogonal $O(p, a_{n})$ Groups

2017 ◽  
Vol 9 (5) ◽  
pp. 37
Author(s):  
ALPER BULUT
Keyword(s):  
Relativistic Theory ◽  
Direct Limit ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Inverse Property ◽  
Finite Dimensional ◽  
Direct Limits ◽  
Bol Loop

A left Bol loop satisfying the automorphic inverse property is called a K-loop or a gyrocommutative gyrogroup. K-loops have been in the centre of attraction since its first discovery by A.A. Ungar in the context of Einstein's 1905 relativistic theory. In this paper some of the infinite dimensional K-loops are built from the direct limit of finite dimensional group transversals.

On Derivations of Lie Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 174-180 ◽  
Author(s):  
Stephen Berman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Characteristic Zero ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Cartan Matrices

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

INFINITE-DIMENSIONAL GENERALIZATIONS OF SIMPLE LIE ALGEBRAS

1990 ◽  
Vol 05 (24) ◽  
pp. 1967-1977 ◽  
Author(s):  
E. S. FRADKIN ◽  
V. YA. LINETSKY
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Virasoro Algebra ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Higher Spin ◽  
Higher Spin Theory

Infinite-dimensional algebras associated with simple finite-dimensional Lie algebra g are considered. Higher-spin generalizations of sl(2) are studied in detail. Those of the Virasoro algebra are viewed as their "analytic continuations". Applications in higher-spin theory and in conformal QFT are discussed.

scholarly journals Pro-Lie groups which are infinite-dimensional Lie groups

2009 ◽  
Vol 146 (2) ◽  
pp. 351-378 ◽  
Author(s):  
K. H. HOFMANN ◽  
K.-H. NEEB
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Exponential Function ◽  
Lie Group ◽  
Smooth Manifold ◽  
Projective Limit ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional Lie Groups

AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

scholarly journals COMPUTATION OF COHOMOLOGY OF LIE SUPERALGEBRAS OF VECTOR FIELDS

2000 ◽  
Vol 11 (02) ◽  
pp. 397-413 ◽  
Author(s):  
V. V. KORNYAK
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Mathematical Physics ◽  
Lie Superalgebras ◽  
Graded Algebras ◽  
Topological Information ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Cohomology Of Lie Algebras ◽  
Mathematics And Physics

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental ("topological") information about algebra under consideration. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why application of computer algebra methods is important for this problem. We describe here an algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras. The program can proceed finite-dimensional algebras and infinite-dimensional graded algebras with finite-dimensional homogeneous components. Among the last algebras, Lie algebras and superalgebras of formal vector fields are most important. We present some results of computation of cohomology for Lie superalgebras of Buttin vector fields and related algebras. These algebras being super-analogs of Poisson and Hamiltonian algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics.

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