Lowest Weight Representations Research Articles

Lowest weight representations of the Schrödinger algebra and generalized heat/Schrödinger equations

The present paper contains two interrelated developments. First, the basic elements of the theory of lowest weight modules, in particular, of Verma modules, over certain non-semisimple Lie algebras are developed in analogy with the semisimple case. This is done on the example of the (central extension of the) Schrödinger algebra in ( n + 1)-dimensional space-time. In more detail is considered the Schrödinger algebra S and its central extension S ̂ in the case n = 1. In particular, there are constructed the singular vectors of S ̂ and the Shapovalov form. The classification of the irreducible lowest weight modules over S ̂ is given. The second development is the proposal of an infinite hierarchy of differential equations, invariant with respect to S ̂ , which are called generalized heat/Schrödinger equations. The ordinary heat/Schrödinger equation is the first member of this hierarchy. These equations are obtained using a vector field realization of S ̂ which provides a polynomial basis realization of the irreducible lowest modules. In some cases the irreducible lowest weight modules are obtained as solution spaces of these differential equations.

Aq-Schrödinger algebra, its lowest-weight representations and generalizedq-deformed heat/Schrödinger equations

We construct a q-deformation of the centrally-extended Schrödinger algebra and an algebraic representation theory through lowest-weight representations. We use Verma modules over , calculate their singular vectors and factorize the Verma modules by submodules built on the singular vectors. We also give a realization of with q-difference operators and obtain a polynomial realization of the lowest-weight representations and an infinite family of q-difference equations which may be called generalized q-deformed heat/Schrödinger equations. We also apply our methods to the on-shell q-Schrödinger algebra proposed by Floreanini and Vinet.

Weight sets of irreducible integrable representations of kac-moody algebras of indefinite type

In this paper, the weight sets of some irreducible and integrable representations, which are not highest or lowest weight representations, of rank two Kac-Moody algebras of indefinite type are completely determined.

Dynamical symmetries inq-deformed quantum mechanics

The dynamical algebra of theq-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction Hamiltonian in terms of the usual canonical variables. Furthermore we construct a welldefined algebraSU q(1,1) with consistent conjugation properties and comultiplication. We obtain non lowest weight representations of this algebra.

Fock Representations of Lie Algebras, Loop Algebras and Affine Lie Algebras11The project supported by National Natural Science Foundation of China.

Explicit Fock representations of the classical Lie algebras in terms of boson creation and annihilation operators with an arbitrary highest weight are derived, and a general rule to construct Fock representations of a loop algebra from a boson realization of its corresponding Lie algebra is established. A new kind of lowest weight representations of the affine Lie algebras attached to the classical Lie algebras, which require a zero center, is also presented. Based on these, a simple affinization procedure is given to obtain the Fock representations of level 1 of these affine Lie algebras.

Continuous spins in 2D gravity: Chiral vertex operators and local fields

We construct the exponentials of the Liouville field with continuous powers within the operator approach. Their chiral decomposition is realized using the explicit Coulomb-gas operators we introduced earlier. From the quantum group viewpoint, they are related to semi-infinite highest- or lowest-weight representations with continuous spins. The Liouville field itself is defined, and the canonical commutation relations are verified, as well as the validity of the quantum Lioville field equations. In a second part, both screening charges are considered. The braiding of the chiral components is derived and shown to agree with the ansatz of a parallel paper of Gervais and Roussel: for continuous spins the quantum group structure U q(sl(2))⊙U q̂(sl(2)) is a nontrivial extension of U q(sl(2)) and U q̂(sl(2)). We construct the corresponding generalized exponentials and the generalized Liouville field.

QUASI-RATIONAL FUSION PRODUCTS

Fusion is defined for arbitrary lowest weight representations of W-algebras, without assuming rationality. Explicit algorithms are given. A category of quasi-rational representations is defined and shown to be stable under fusion. Conjecturally, it may coincide with the category of representations of finite quantum dimensions.

REPRESENTATIONS OF THE SN-EXTENDED HEISENBERG ALGEBRA AND RELATIONS BETWEEN KNIZHNIK-ZAMOLODCHIKOV EQUATIONS AND QUANTUM CALOGERO MODEL

We discuss lowest weight representations of the SN-extendedHeisenberg algebras underlying the N-body quantum-mechanical Calogero model. Our construction leads to flat derivatives interpolating between Knizhnik-Zamolodchikov and Dunkl derivatives. It is argued that based on these results one can establish new links between solutions of the Knizhnik-Zamolodchikov equations and wave functions of the Calogero model.

W-Algebras, new rational models and completeness of thec=1 classification

Two series ofW with two generators are constructed from chiral vertex operators of a free field representation. Ifc=1–24k, there exists aW(2, 3k) algebra for k ∈ ℤ+/2 and aW(2, 8k) algebra for k ∈ ℤ+/4. All possible lowest-weight representations, their characters and fusion rules are calculated proving that these theories are rational. It is shown, that these non-unitary theories complete the classification of all rational theories with effective central chargec eff=1. The results are generalized to the case of extended supersymmetric conformal algebras.

FUSION RULES FOR THE FRACTIONAL LEVEL $\widehat{{\rm sl}(2)}$ ALGEBRA

We derive, as the condition for the null vector decoupling, the fusion rules for the [Formula: see text] algebra with fractional level, which have an interesting structure related to affine Weyl transformation. It is shown that, to get non-trivial fusion rules, we must include the primary field which belongs to neither the highest nor the lowest weight representations.

Quantum group extended chiral p-models

The quantum symmetry group U q of an extended chiral conformal model is determined by the requirement that symmetry transformations commute with braid group statistics operators and by the relation between fusion rules and tensor product expansions of a certain class of U 4 representations. For thermal minimal “ p-models”, involving no more than p − 1 unitary lowest weight representations of the Virasoro algebra Vir, U 4 is the quantum universal enveloping (QUE) algebra U 4(sl(2)) with deformation parameter q satisfying q + q −1 = 2 cos π/ p ( q p = − 1, p = 4, 5,…). To each 2-dimensional local field labelled by a pair of nonnegative integers v, v ̄ (0 ⩽ v, v ̄ ⩽ p − 2) we make correspond an analytic chiral field φ v , of weight Δ v and q- spin I v ̄ . The correlation functions of φ v , transform under an 1-dimensional unitary representation of the braid group. As a result we reproduce the ADE classification of 2-dimensional p models in terms of their extended chiral counterparts. It turns out that U q -extended chiral p-models always involve non-unitary and indecomposable representations of Vir.

Unitary lowest weight representations of the noncompact supergroup OSp(2m*/2n)

The oscillator construction of the unitary lowest weight representations of the noncompact supergroup OSp(2m*/2n) with the even subgroup SO*(2m)×USp(2n) is given. Simple rules for determining the SO*(2m)×USp(2n) decomposition of the lowest weight representations of OSp(2m*/2n) are derived. The particular examples of OSp(4*/2) with the even subgroup SU(1,1)×SU(2)×SU(2) and OSp(4*/6) with the even subgroup SU(1,1)×SU(2)×Usp(6) are studied in greater detail.

Lowest weight representations of some infinite dimensional groups on fock spaces

The results of Kashiwara and Vergne on the decomposition of the tensor products of the ‘Segal-Shale-Weil representation’ are extended to the infinite dimensional case and give all unitary lowest weight representations. Our methods are basically algebraic. When restricted to the finite dimensional case, they yield a new proof.

Unitary lowest weight representations of the noncompact supergroup OSp(2n/2m,R)

The oscillator construction of the unitary irreducible lowest (highest) weight representations of the noncompact supergroup OSp(2n/2m,R) with even subgroup SO(2n)×Sp(2m,R) is given. In particular, simple rules for determining the SO(2n)×Sp(2m,R) decomposition of the unitary lowest weight representations of OSp(2n/2m,R) are derived.

The spectrum of Wess-Zumino-Witten models

We consider the SU(2) and SO(3) Wess-Zumino-Witten models in the Schrödinger representation. Wave functions are sections of a line bundle over a loop group. We compute the spectrum of irreducible lowest weight representations appearing in the model. Fusion rules of the operator product expansion are also discussed.

An approach to BRST formulation of Kac-Moosy algebra

Based on an ordinary formulation of BRST the BRST charge concerning the affine Kac-Moody algebra g with the central extension is investigated. The nilpotency is satisfied at the value of central charge = -(quadratic Casimir of Ad ( G ), which shows the theory to fall into the unphysical lowest weight representation sector without lower bound in energy. No improvement is made by a semi-direct coupling to the Virasoro algebra, in which the central charge of the Virasoro algebra gets a modification: C − 26 − 2 dim G .

Coherent state theory of the noncompact symplectic group

An extended coherent state theory is presented for the noncompact Sp(3,R) group which reveals a simple relationship between the Sp(3,R) algebra and its contracted u(3)-boson limit. The relationship is used to derive a remarkably accurate analytic expression for Sp(3,R) matrix elements for the generic lowest-weight representations. The expression is shown to be exact whenever the states involved are multiplicity free with respect to the u(3) subalgebra. It is further shown how exact matrix elements are easily calculated in general. Dyson and Holstein–Primakoff type u(3)-boson expansions are given.

Analytical expressions for the matrix elements of the non-compact symplectic algebra

Analytic matrix elements are derived for all the lowest weight representations of the noncompact symplectic algebra sp(3,R). It is shown that the expressions are exact for representations of the type ( sigma 1= sigma 2= sigma 3) and for all states of an arbitrary representation that are multiplicity free with respect to the u(3) subalgebra. Furthermore they are remarkably accurate in general.