The irreducible highest weight representations ofc∞

Let G be a connected noncompact simple Lie group with finite center and let A be a maximal compact subgroup of G. Assume the space G/K is Hermitian symmetric. We associate to each irreducible representation τ of K a principal series representation W(τ) and a (7-equivariant Szegό-type integral operator Sτ such that Sτ maps the K-finite vectors in W(τ) onto an irreducible highest weight $module L(τ). Of primary concern here are those representations τ which are reduction points. For such τ, we construct certain systems 3lτ of ΰ-equivariant differential operators and then utilize 3fτ to establish the infinitesimal irreducibility of the image of Sτ. 1. Introduction. Let G be a connected noncompact simple Lie group with finite center and let K be a maximal compact subgroup of G. Assume the space G/K is Hermitian symmetric. The main purpose of this article is to realize each irreducible highest weight representation of G as the image of a G-equivariant quotient map defined on principal series representations. To make this more precise, recall that each irreducible highest weight representation πτ of G is parametrized by an irreducible unitary representation τ of K. Let C°°(G, τ) denote the space of τ-covariant C°°-functions on G. We associate to τ a principal series representation W(τ) and a Szego map Sτ: W(τ) -> C°°(G, τ) having the property that the ^-finite vectors in W(τ) are mapped onto an irreducible g-module equivalent to the derived action of πτ. In the case of discrete series and limits thereof, this type of result was proved by Knapp and Wallach [16] in the general setting where G is a semisimple equirank Lie group with finite center. The main result here is that the irreducibility of the image of *S τ persists for all highest weight representations. Moreover, for certain τ called reduction points, the irreducibility of Image (Sτ) is proved by showing this space is annihilated by a system Qίτ of Gequivariant differential operators. The system 3τ somewhat parallels the role of the Schmid operator in the Knapp and Wallach result. The realization of distinguished representations as irreducible images of quotient maps is a recurring theme in the literature which

The highest weight representation of the infinite Lie algebras is used to construct various soliton equations by the famous Kyoto school. In this chapter, we firstly review the construction of the BKP hierarchy corresponding to o(∞) by using the neutral free Fermions. As an important reduction of the BKP hierarchy, the constrained BKP hierarchy is further investigated in the aspect of the bilinear equations. It is shown that the bilinear equations in the form of tau functions can fully determine the constrained BKP hierarchy. Then the bilinear equations of the constrained BKP hierarchy are given in the Fermionic picture, and further interpretations of these bilinear equations are obtained. Particularly, there is the structure of the modified BKP hierarchy in the constrained BKP hierarchy, which is presented in the forms of the bilinear equations. Last, we investigate the dressing structure and the Lax structure of the modified BKP hierarchy. Note that the soliton equations obtained by the highest weight representation are usually given in the forms of the bilinear equations; therefore, the work here may be one try in further study of the integrable properties for these soliton equations.

abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.

We construct and study in detail various dual pairs acting on some infinite dimensional Fock representations between a finite dimensional classical Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to the action of a dual pair, present explicit formulas for the highest weight vectors and calculate the corresponding highest weights. We further outline applications of these dual pairs to the study of tensor products of modules of such an infinite dimensional Lie algebra.

Supersymmetry and parastatistics

Projective modules over classical Lie algebras of infinite rank in the parabolic category

We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For S L ( r ) SL(r) the irreducibility result is the following: let λ = ( a 1 ≥ a 2 ≥ ⋯ ≥ a r − 1 ≥ 0 ) \lambda =(a_1\geq a_2\geq \cdots \geq a_{r-1}\geq 0) be the highest weight of an irreducible rational representation V λ V_{\lambda } of S L ( r ) SL(r) . Assume that the integers a 1 + r − 1 , a 2 + r − 2 , ⋯ , a r − 1 + 1 a_1+r-1, ~a_2+r-2, \cdots , a_{r-1}+1 are relatively prime. Then the character χ λ \chi _{\lambda } of V λ V_{\lambda } is strongly irreducible in the following sense: for any natural number d d , the function χ λ ( g d ) , g ∈ S L ( r , C ) \chi _{\lambda }(g^d), ~g\in SL(r,\mathbb {C}) is irreducible in the ring of regular functions of S L ( r , C ) SL(r,\mathbb {C}) .

Motivated by an example of Inönü and Wigner, a general procedure for the contraction of irreducible unitary representations of semisimple compact Lie algebras is given. In this situation, contractions mean generalized Inönü–Wigner contractions. In particular, this means that by a simple limiting process, one obtains representations of nonsemisimple noncompact Lie algebras. As examples, a general treatment of SO(3) is given as well as some results for SU(3).

It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.

Families of operator identities appeared as a consequence of an existence of finite-dimensional representation of (super) Lie algebras of first-order differential operators and $q$-deformed (quantum) algebras of first-order finite-difference operators are presented. It is shown that those identities can be rewritten in terms of creation/annihilation operators and it leads to a simplification of the problem of the normal ordering in the second quantization method.

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi–Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.

Representations of twisted multi-loop Lie algebras

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.

Polynomial identities satisfied by the infinitesimal generators of a semi-simple Lie group are employed to construct projection operators which project a tensor product representation V( lambda )(X)V (V( lambda ) finite dimensional and irreducible, V an infinite dimensional representation admitting an infinitesimal character) onto a primary subrepresentation. Such projection operators project out the generalised shift components of a tensor operator acting on an infinite dimensional representation V in direct analogy with the finite dimensional analysis of Bracken and Green (1971). Applications of these methods to evaluating the matrix elements of the generators of the locally compact groups U(p,q) and O(p,q) in unitary (discrete) irreducible representations are discussed.

Killing – An algebraic computational package for Lie algebras