Modular Lie Algebras and their Representations

The basic structure and representation theory of finite-dimensional simple Lie algebras over \(\mathbb{C} \), as developed by W. Killing, E. Cartan and H. Weyl, is a well-known and frequently used branch of mathematics. It is a striking fact that very significant new results in this classical area of mathematics have come to light during the past decade. These new results have originated from the development of the theory of quantum groups. The purpose of this article is to survey briefly the classical results on the representation theory of simple Lie algebras and then to describe the recent developments primarily due to G. Lusztig, M. Kashiwara and P. Littelmann.

Various methods of representation theory of modular Lie algebras are improved. As an application the structure of the Lie algebras having a faithful irreducible module of dimension > p 2 > {p^2} is determined. Applications to the classification theory of modular simple Lie algebras are given.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

One fruitful motivating principle of much research on the family of integrable systems known as “Toda lattices” has been the heuristic assumption that the periodic Toda lattice in an affine Lie algebra is directly analogous to the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper shows that the analogy is not perfect. A discrepancy arises because the natural generalization of the structure theory of finite-dimensional simple Lie algebras is not the structure theory of loop algebras but the structure theory of affine Kac-Moody algebras. In this paper we use this natural generalization to construct the natural analog of the nonperiodic Toda lattice. Surprisingly, the result is not the periodic Toda lattice but a new completely integrable system on the periodic Toda lattice phase space. This integrable system is prescribed purely in terms of Lie-theoretic data. The commuting functions are precisely the gauge-invariant functions one obtains by viewing elements of the loop algebra as connections on a bundle over S.

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

This paper is a contribution to the structure and representation theory of finitedimensional cylindric and polyadic equality algebras.The main result is that every singulary algebra is representable, where an algebra is singular y if it is generated by elements supported by singletons.In particular, all prime algebras are representable (a prime algebra has no proper subalgebra, and every cylindric or polyadic equality algebra has a prime subalgebra).Since every polyadic algebra can be embedded in a polyadic equality algebra the results also apply to them.Every infinite-dimensional singulary algebra is locally finite and hence is known to be representable, so we shall be concerned just with the finite-dimensional case, where previously very little was known about representation.It may be mentioned that for any dimension greater than one there are nonrepresentable cylindric and polyadic equality algebras(2).For some special kinds of singulary algebras we obtain a direct construction of the representation, which brings out the structure of the algebra clearly.The methods used essentially constitute an algebraic version of Behmann's solution of the decision problem for the singulary predicate calculus with equality(3).Since only finitely many variables are available, the algebraization has some novel features.The essential ideas can be seen in the case of prime algebras, for which no advanced results about cylindric or polyadic algebras are needed.In the general case, however, use is made of the fact that a cylindric algebra is representable if every finitely generated subalgebra of it is.In the last section of the paper we discuss the logical counterpart of the methods and results of the algebraic part of the paper.1. Introduction.A cylindric algebra of dimension a (a CAa) is a system 21 = <[A, +,-,-, cK,dKXyKtX<x such that (A, +,-, -> is a Boolean algebra, a is an ordinal, cK is a singulary Presented to the Society, November 17,1962 under the title On finite dimensional cylindric algebras; received by the editors April 2, 1963.0) The research herein reported on was supported by

The algebraic structure and Poisson's integral theory of mechanico-electrical systems are studied. The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic forms for mechanico-electrical systems are obtained. The Lie algebraic structure and the Poisson's integral theory of Lagrange mechanico-electrical systems are derived. The Lie algebraic structure admitted and Poisson's integral theory of the Lagrange?Maxwell mechanico-electrical systems are presented. Two examples are presented to illustrate these results.

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

Since its inception in the early seventies, the study of Enveloping Algebras has undergone a significant and continuous evolution and moreover has inspired a wide variety of developments in many areas of mathematics including Ring Theory, Differential Operators, Invariant Theory, Quantum Groups and Hecke Algebras. As indicated above, one of the main goals behind this meeting was to bring together a group of participants with a wide range of interests in and around the geometric and the combinatorial side of the representation theory of Lie groups and algebras. We strongly believe that such an approach to representation theory, in particular interaction between geometry and representation theory, will open up new avenues of thought and lead to progress in a number of areas. This diversity was well reflected in the expertize represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under the following three headings: (1) The study of the structure and representation theory through a) Slodowy slices and the Joseph ideal, b) Equivalence of categories of representations and certain geometric counterparts, c) Hall algebras, Quotient schemes and canonical bases, d) Richardson elements and birationality questions, e) free Lie algebras and current algebras, f) Dirac cohomology, h) Gelfand-Zeitlin theory (2) Combinatorial Aspects of Lie Systems particularly through a) Saturation problems and Buildings, b) Affine and double Affine Hecke algebras, Cherednik algebras, c) Solvable lattice models (3) Geometric Structures including a) Representation theoretic methods in enumerative geometry, b) Multiplicity free symplectic reduction, c) Monodromy actions of braid groups, d) Associated varieties for Lie super algebras, e) affine Grassmannians, f) Schubert varieties and Demazure modules The single most important development reported in this conference seems to us to be the generalization of the categorical Satake isomorphism to quantum groups at roots of unity, which allows to bring in a whole lot of new geometry into representation theory. It seems likely that it will lead to the proofs of some open conjectures of Lusztig. We were particularly happy to have quite a few very strong younger participants and some highly promising postdocs. We tried to accomodate young postdocs, as much as feasible, to present their work by devoting one afternoon to their (shorter) talks.

We provide a new approach towards the analysis of the fusion products defined by B. Feigin and S. Loktev in the representation theory of (truncated) current Lie algebras. We understand the fusion product as a degeneration using Gröbner theory of non-commutative algebras and outline a strategy on how to prove a conjecture about the defining relations for the fusion product of two evaluation modules. We conclude with following this strategy for mathfrak {sl}_2(mathbb {C}[t]) and hence provide yet another proof for the conjecture in this case.

Representations of Lie groups are abstractions of continuous symmetries of linear spaces.By considering their differentials (linear approximations), we obtain representations of Lie algebras.These are regarded as generalizations of the Fourier analysis and important tools not only in mathematics but also in physics.In this article, I present fundamental and important examples on representations of Lie groups and Lie algebras and present some of my latest results.

CHAPTER 13 - POINCARÉ-BIRKHOFF-WITT THEOREM AND ITS APPLICATIONS TO REPRESENTATION THEORY OF SEMISIMPLE LIE ALGEBRAS

2-Variable generalized hermite matrix polynomials and lie algebra representation

The representation of symmetry groups plays a crucial role in physics. In this chapter we discuss the elements of the representation theory of Lie groups and Lie algebras. In particular, we apply representations of the Lie group SU(3) and the Lie algebra su(3) to the quark model in strong interaction.