Irreducible Representations Of Lie Algebra Research Articles

Dimensions of modular irreducible representations of semisimple Lie algebras

In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished p p -characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category O \mathcal {O} . For this, we prove an equivalence between two categorifications of a parabolically induced module over the affine Hecke algebra conjectured by the first named author. For the general nilpotent p p -character, we get character formulas by explicitly computing the duality operator on a suitable equivariant K-group.

Classification of irreducible representations of Lie algebra of vector fields on a torus

Abstract We solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on n-dimensional torus for any n. This generalizes the classical result of O.Mathieu on simple weight modules for the Virasoro algebra ( n = 1 ${n=1}$ ). Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao.

Kravchuk oscillator revisited

The study of irreducible representations of Lie algebras and groups has traditionally considered their action on functions of a continuous manifold (e.g. the 'rotation' Lie algebra so(3) on functions on the sphere). Here we argue that functions of a discrete variable -Kravchuk functions- are on equal footing for that study in the case of so(3). They lead to a discrete quantum model of the harmonic oscillator, and offer a corresponding set of special function relations. The technique is applicable to other special function families of a discrete variable, which stem from low-dimensional Lie algebras and are stationary solutions for the corresponding discrete quantum models.

CleGo: A package for automated computation of Clebsch–Gordan coefficients in tensor product representations for Lie algebras [formula omitted

We present a program that allows for the computation of tensor products of irreducible representations of Lie algebras A–G based on the explicit construction of weight states. This straightforward approach (which is slower and more memory-consumptive than the standard methods to just calculate dimensions of the tensor product decomposition) produces Clebsch–Gordan coefficients that are of interest for instance in discussing symmetry breaking in model building for grand unified theories. For that purpose, multiple tensor products have been implemented as well as means for analyzing the resulting effective operators in particle physics. Program summaryProgram title: CleGoCatalogue identifier: AEIQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 3641No. of bytes in distributed program, including test data, etc.: 34 536Distribution format: tar.gzProgramming language: OʼCamlComputer: i386–i686, x86_64Operating system: Cross-platform, for definiteness though we assume some UNIX system.RAM: ⩾4 GB commendable, though in general memory requirements depend on the size of the Lie algebras and the representations involved.Classification: 4.2, 11.1Nature of problem: Clebsch–Gordan coefficients are widely used in physics. This program has been written as a means to analyze symmetry breaking in the context of grand unified theories in particle physics. As an example, we computed the singlets appearing in higher-dimensional operators 27⊗27⊗27⊗78 and 27⊗27⊗27⊗650 in an E6-symmetric GUT.Solution method: In contrast to very efficient algorithms that also produce tensor product decompositions (as far as outer multiplicities/Littlewood–Richardson coefficients are concerned) we proceed straightforwardly by constructing all the weight states, i.e. the Clebsch–Gordan coefficients. This obviously comes at the expense of high memory and CPU-time demands. Applying Dynkin arithmetic in weight space, the algorithm is an extension of the one for the addition of angular momenta in su(2)≈A1, for reference see [1]. Note that, in general, Clebsch–Gordan coefficients are basis-dependent and therefore need to be understood with respect to the chosen basis. However, singlets appearing in (multiple) tensor products are less basis-dependent.Restrictions: Generically, only tensor products of non-degenerate or adjoint representations can be computed. However, the irreps appearing therein can subsequently be used as new input irreps for further tensor product decomposition so in principle there is no restriction on the irreps the tensor product is taken of. In practice, though, it is by the very nature of the explicit algorithm that input is restricted by memory and CPU runtime requirements.Unusual features: Analytic computation instead of float numerics.Additional comments: The program can be used in “notebook style” using a suitable OʼCaml toplevel. Alternatively, an OʼCaml input file can be compiled which results in processing that is approximately a factor of five faster. The latter mode is commendable when large irreps need to be constructed.Running time: Varies depending on the input from parts of seconds to weeks for very large representations (because of memory exhaustion).Reference:[1]J. Fuchs, C. Schweigert, Symmetries, Lie Algebras and Representations, Cambridge Univ. Press, Cambridge, UK, 1997, 438 p.

An explicit construction of Casimir operators and eigenvalues. II

We provide a way of computing Casimir eigenvalues for Weyl orbits as well as for irreducible representations of Lie algebras. A κ(s) number of polynomials of rank N are obtained explicitly for AN Casimir operators of order s where κ(s) is the number of partitions of s into positive integers except 1. It is also emphasized that these eigenvalue polynomials prove useful in obtaining formulas to calculate weight multiplicities and in explicit calculations of the whole cohomology ring of classical and also exceptional Lie algebras.

Relation between the irreducible representations of Lie algebras and the irreducible representations of p-groups

A proof is given of a theorem stating that there is a correspondence between the irreducible complex representations of a finite p-group and the irreducible representations of its associated nilpotent Lie algebra over a field of characteristic p. As a corollary it is found that the sets of degrees of the irreducible representations are the same.

Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture

Letg be the Lie algebra of a connected reductive groupG over an algebraically closed field of characteristicp>0. Suppose thatG(1) is simply connected andp is good for the root system ofG. Ifp=2, suppose in addition thatg admits a nondegenerateG-invariant trace form. LetV be an irreducible and faithfulg-module withp-character χ∈g*. It is proved in the paper that dimV is divisible byp1/2dimΩ(χ) where Ω(χ) stands for the orbit of χ under the coadjoint action ofG.

Existence of identities in the restricted enveloping algebra

By the Latyshev theorem, the universal enveloping algebra of a Lie algebra over a field of zero characteristic is a PI-algebra if and only if this Lie algebra is abelian. The Bakhturin theorem holds for each Lie algebra G of positive characteristic: The universal enveloping algebra has a nontrivial identity if and only if G is almost abelian and all inner derivations adx, x ~ G, are algebraic of bounded degree, depending only on G. This result was used by Bakhturin for the study of dimensions of irreducible representations of Lie algebras of positive characteristic. The proofs of these results are given in [i, Chap. 6] (see also [2-4]).

Resonance widths and positions by an algebraic approach.

The use of nonunitary discrete, irreducible representations of Lie algebras is proposed as a method for the determination of widths and energies of resonances. The method is illustrated by explicit application to three kinds of resonances: in transmission over a well, in transmission over a barrier, and in barrier penetration for a family of one-dimensional potentials. The same procedure is also useful in an algebraic approach to scattering by a complex (optical) potential, and for parametrizing of the resonant contribution to the scattering matrix.

A remark on irreducible representations of Lie algebras

AbstractIn addition to the results of the paper (Bachturin (1974)) we give the precise form of the necessary and sufficient conditions ensuring that all irreducible representations of a Lie algebra were of finite bounded degree.

Step algebras of semi-simple subalgebras of Lie algebras

For each pair ( G, K) where G is a complex finite-dimensional Lie algebra and K a semi-simple subalgebra of G, we construct an associative algebra (step algebra) Y ( G, K) and a homomorphism i *: Y ( G, K)→ E( G) is the enveloping algebra of G. Y ( G, K) has the following properties: (1) If V is any G-module and x ϵ V a K-maximal vector, then sx = i * ( s) x is K-maximal for any s ϵ Y ( G, K); (2) If V is irreducible and a certain simple criteria is fulfilled, then any K-maximal vector can be written in the form sx m , s ϵ Y ( G, K), where x m is some fixed K-maximal vector. Because of these properties Y ( G, K) has great practical value when constructing irreducible representations of Lie algebras in a form which makes the reduction with respect to a semi-simple subalgebra explicit.

Representations Subduced on an Ideal of a Lie Algebra

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

On Real Irreducible Representations of Lie Algebras

Let us consider the following two problems:Problem A. Let g be a given Lie algebra over the real number field R. Then find all real, irreducible representations of g.Problem B. Let n be a given positive integer. Then find all irreducible subalgebras of the Lie algebra ôí(w, R) of all real matrices of degree n.