Representations Of Lie Algebras Research Articles

Representation theory of $\mathcal{W}$ -algebras

We study the representation theory of the $\mathcal{W}$ -algebra $\mathcal{W}_k(\bar{\mathfrak{g}})$ associated with a simple Lie algebra $\bar{\mathfrak{g}}$ at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of $\mathcal{W}_k(\bar{\mathfrak{g}})$ is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra $\mathfrak{g}$ of $\bar{\mathfrak{g}}$ . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of $\mathcal{W}$ -algebras.

Generating Relations of Hermite–Tricomi Functions Using a Representation of Lie Algebra 𝑇3

Abstract Motivated by recent studies of the properties of new classes of polynomials constructed in terms of quasi-monomials, certain generating relations involving Hermite–Tricomi functions are obtained. To accomplish this we use the representation 𝑄(𝑤, 𝑚0) of the 3-dimensional Lie algebra 𝑇3. Some special cases are also discussed.

Representations of the Twisted Heisenberg–Virasoro Algebra and the Full Toroidal Lie Algebras

An explicit construction of indecomposable modules for the twisted Heisenberg–Virasoro algebra and representations for the full toroidal Lie algebras are given.

Quantum Super-Integrable Systems as Exactly Solvable Models

We consider some examples of quantum super-integrable systems and the as- sociated nonlinear extensions of Lie algebras. The intimate relationship between super- integrability and exact solvability is illustrated. Eigenfunctions are constructed through the action of the commuting operators. Finite dimensional representations of the quadratic al- gebras are thus constructed in a way analogous to that of the highest weight representations of Lie algebras.

Associative homotopy Lie algebras and Wronskians

We analyze representations of Schlessinger-Stasheff associative homotopy Lie algebras by higher-order differential operators. W-transformations of chiral embeddings of a complex curve related with the Toda equations into Kahler manifolds are shown to be endowed with the homotopy Lie-algebra structures. Extensions of the Wronskian determinants preserving Schlessinger-Stasheff algebras are constructed for the case of n ≥ 1 independent variables.

Thin coverings of modules

Thin coverings are a method of constructing graded-simple modules from simple (ungraded) modules. After a general discussion, we classify the thin coverings of (quasifinite) simple modules over associative algebras graded by finite abelian groups. The classification uses the representation theory of cyclotomic quantum tori. We close with an application to representations of multiloop Lie algebras.

Structure of the Enveloping Algebras

The adjoint representations of several small dimensional Lie algebras on their universal enveloping algebras are explicitly decomposed. It is shown that commutants of raising operators are generated as polynomials in several basic elements. The explicit form of these elements is given and the general method for obtaining these elements is described.

Multiplicity-free branching rules for outer automorphisms of simple Lie algebras

We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras 𝔤 to the fixed point subalgebras 𝔤 σ of outer automorphisms σ . The representations have highest weights which are scalar multiples of fundamental weights or linear combinations of two scalar ones. Our list of pairs of Lie algebras ( 𝔤 , 𝔤 σ ) includes an exceptional symmetric pair ( E 6 , F 4 ) and also a non-symmetric pair ( D 4 , G 2 ) as well as a number of classical symmetric pairs. Some of the branching rules were known and others are new, but all the rules in this paper are proved by a unified method. Our key lemma is a characterization of the ``middle'' cosets of the Weyl group of 𝔤 in terms of the subalgebras 𝔤 σ on one hand, and the length function on the other hand.

Exploring Representation Theory of Unitary Groups via Linear Optical Passive Devices

In this paper, we investigate some mathematical structures underlying the physics of linear optical passive (LOP) devices. We show, in particular, that with the class of LOP transformations on N optical modes one can associate a unitary representation of U (N) in the N-mode Fock space, representation which can be decomposed into irreducible sub-representations living in the subspaces characterized by a fixed number of photons. These (sub-)representations can be classified using the theory of representations of semi-simple Lie algebras. The remarkable case where N = 3 is studied in detail.

Representations of toroidal extended affine Lie algebras

We show that the representation theory for the toroidal extended affine Lie algebra is controlled by a VOA which is a tensor product of four VOAs: a sub-VOA V Hyp + of a hyperbolic lattice VOA, affine g ˙ ˆ and sl ˆ N VOAs and a Virasoro VOA. A tensor product of irreducible modules for these VOAs admits the structure of an irreducible module for the toroidal extended affine Lie algebra. We also show that for N = 12 , V Hyp + becomes an exceptional irreducible module for the extended affine Lie algebra of rank 0.

Extremal Bases for the Adjoint Representations of the Simple Lie Algebras

We construct n distinct weight bases, which we call extremal bases, for the adjoint representation of each simple Lie algebra 𝔤 of rank n: One construction for each simple root. We explicitly describe actions of the Chevalley generators on the basis elements. We show that these extremal bases are distinguished by their “supporting graphs” in three ways. (In general, the supporting graph of a weight basis for a representation of a semisimple Lie algebra is a directed graph with colored edges that describe the supports of the actions of the Chevalley generators on the elements of the basis.) We show that each extremal basis constructed is essentially the only basis with its supporting graph (i.e., each extremal basis is solitary), and that each supporting graph is a modular lattice. Each extremal basis is shown to be edge-minimizing: Its supporting graph has the minimum number of edges. The extremal bases are shown to be the only edge-minimizing as well as the only modular lattice weight bases (up to scalar multiples) for the adjoint representation of 𝔤. The supporting graph for an extremal basis is shown to be a distributive lattice if and only if the associated simple root corresponds to an end node for a “branchless” simple Lie algebra, i.e., type A, B, C, F, or G. For each extremal basis, basis elements for the Cartan subalgebra are explicitly expressed in terms of the h i Chevalley generators.

SMALL MAXIMAL SPACES OF NON-INVERTIBLE MATRICES

The rank of a vector space A of n × n-matrices is by definition the maximal rank of an element of A. The space A is called rank-critical if any matrix space that properly contains A has a strictly higher rank. This paper exhibits a sufficient condition for rank-criticality, which is then used to prove that the images of certain Lie algebra representations are rank-critical. A rather counter-intuitive consequence, and the main novelty in this paper, is that for infinitely many n, there exists an eight-dimensional rank-critical space of n × n-matrices having generic rank n − 1, or, in other words: an eight-dimensional maximal space of non-invertible matrices. This settles the question, posed by Fillmore, Laurie, and Radjavi in 1985, of whether such a maximal space can have dimension smaller than n. Another consequence is that the image of the adjoint representation of any semisimple Lie algebra is rank-critical; in both results, the ground field is assumed to have characteristic zero. 2000 Mathematics Subject Classification 15A30, 17B10, 20G05.

Representations and Cocycle Twists of Color Lie Algebras

We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra \({\text{sl}}^{c}_{2} \).

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Abstract In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

On the Structure of a Class of Graded Modules Related to Symmetric Pairs

Let [Formula: see text] be the Cartan decomposition of a real semisimple Lie algebra and [Formula: see text] be its complexification. Let [Formula: see text] be the corresponding isotropy representation, and the exterior algebra [Formula: see text] becomes a graded [Formula: see text]-module by extending ν. We study a graded [Formula: see text]-submodule C of [Formula: see text] and get two important decompositions of the [Formula: see text]-module [Formula: see text]. Let [Formula: see text] be the symmetric algebra over [Formula: see text]. Then [Formula: see text] also has an [Formula: see text]-module structure, which is [Formula: see text]-equivariant, and C is a space of generators for this module. Our results generalize Kostant's results in the special case that ν is the adjoint representation of a semisimple Lie algebra.

Bäcklund transformations for new integrable hierarchies related to the polynomial Lie algebra [formula omitted

In a recent paper [P. Casati, G. Ortenzi, New integrable hierarchies from vertex operator representations of polynomial Lie algebras, J. Geom. Phys. 56 (3) (2006) 418–449] Casati and Ortenzi gave a representation-theoretic interpretation of recently discovered coupled soliton equations, which were described by e.g. R. Hirota, X. Hu, X. Tang [A vector potential KdV equation and vector Ito equation: Soliton solutions, bilinear Bäcklund transformations and Lax pairs, J. Math. Anal. Appl. 288 (1) (2003) 326–348. [3]], S. Kakei [Dressing method and the coupled KP hierarchy, Phys. Lett. A 264 (6) (2000) 449–458. [6]] and S.Yu. Sakovich [A note in the Painlevé property of coupled KdV equation, arXiv:nlin.SI/0402004. [7]]. Casati and Ortenzi use vertex operators for these Lie algebras and a boson–fermion type of correspondence to get a hierarchy of coupled Hirota bilinear equations. In this paper we reformulate the Hirota bilinear description for the Lie algebra g l ∞ ( n ) and obtain a bilinear identity for matrix wave functions. From that it is straightforward to deduce the Sato–Wilson, Lax and Zakharov–Shabat equations. Using these wave functions and standard calculus with vertex operators we obtain elementary Bäcklund–Darboux transformations.

Solitary and edge-minimal bases for representations of the simple lie algebra [formula omitted

We consider two families of weight bases for “one-rowed” irreducible representations of the simple Lie algebra G 2 constructed in Donnelly et al [Constructions of representations of o ( 2 n + 1 , C ) that imply Molev and Reiner–Stanton lattices are strongly Sperner, Discrete Math. 263 (2003) 61–79] using two corresponding families of distributive lattices (called “supporting graphs”), here denoted L G 2 LM ( k ) and L G 2 RS ( k ) . We exploit the relationship between these bases and their supporting graphs to give combinatorial proofs that the bases enjoy certain uniqueness and extremal properties (the “solitary” and “edge-minimal” properties, respectively). Our application of the combinatorial technique we develop in this paper to obtain these results relies on special total orderings of the elements and edges of the lattices. We also apply this technique to another family of lattice supporting graphs to re-derive results obtained in Donnely et al. using different, more algebraic methods.

A class of primary representations associated with symmetric pairs and restricted root systems

Let v:r → so(p) be a representation of a complex reductive Lie algebra r on a complex vector space p. Assume that v is the complexified differential of an orthogonal representation of a compact Lie group R. Then the exterior algebra ^p becomes an r-module by extending v. Let Spin v: r → End S be the composition of v with the spin representation Spin::so(p) → End S. We completely classify the representations v for which the corresponding Spin v representation is primary, give a description of the r-module structure of ^p, and present a decomposition of the Clifford algebra over p. It turns out that, if the Spin v representation is primary, v must be an isotropy representation of some symmetric pair. Our work generalizes Kostant's well-known results that dealt with the special case when v is the adjoint representation of a semisimple Lie algebra. In the proof we introduce the restricted root system of a real semisimple Lie algebra, which is of independent interest.

Mullineux involution and twisted affine Lie algebras

We use Naito and Sagaki's work [S. Naito, D. Sagaki, Lakshmibai–Seshadri paths fixed by a diagram automorphism, J. Algebra 245 (2001) 395–412; S. Naito, D. Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Algebra 251 (2002) 461–474] on Lakshmibai–Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type A 2 ℓ ( 2 ) and of type D ℓ + 1 ( 2 ) . We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijections between the set of restricted strict partitions (respectively, the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (respectively, in the even case) are obtained.

Littelmann paths for the basic representation of an affine Lie algebra

A highest-weight representation of an affine Lie algebra g ˆ can be modeled combinatorially in several ways, notably by the semi-infinite paths of the Kyoto school and by Littelmann's finite paths. In this paper, we unify these two models in the case of the basic representation of an untwisted affine algebra, provided the underlying finite-dimensional algebra g possesses a minuscule representation (i.e., g is of classical or E 6 , E 7 type). We apply our “coil model” to prove that the basic representation of g ˆ , when restricted to g , is a semi-infinite tensor product of fundamental representations, and certain of its Demazure modules are finite tensor products.