Finite-dimensional representations of the euclidean group in the plane

In this article two theorems are given which permit, together with the concept of a representation vector diagram, to classify all (linear) finite-dimensional representations of the algebra and group E ₂ which are induced by a master representation on the place of the universal enveloping algebra of the algebra E ₂. Apart from a classification of the finite-dimensional representations, the two theorems make it possible to obtain the matrix elements of these representations for both, algebra and group, in explicit form. The material contained in this letter forms part of an analysis of indecomposable (finite- and infinite-dimensional) representations of the algebra and group E ₂ which is contained in Reference [1]. No proofs will be given in this letter. We refer instead to [1].     

Polynomial associative algebras of quantum superintegrable systems

The integrals of motion of classical two-dimensional superintegrable systems, with polynomial integrals of motion, close in a restrained polynomial Poisson algebra; the general form of the quadratic case is investigated. The polynomial Poisson algebra of the classical system is deformed into a quantum associative algebra of the corresponding quantum system, and the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. The finite-dimensional representations of the algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the roots of algebraic equations in the quadratic case.     

Superfields as Representations

We study irreducible and reducible representations of the generalized Lie algebra of Wess and Zumino. The algebra is integrated to a group with the help of Grassmann algebras and the representations of the algebra are made into representations of the group. We construct invariant sesquilinear forms that are positive definite for the Wess-Zumino algebra over the complex field. We define irreducible superfields for any spin J as specific realizations of such representations. All superfields appearing in the literature are either equivalent to one of these or built up out of these superfields.     

On conformal invariance of interacting fields

We study the action of the conformal algebra on interacting fields. On a certain set of states the algebra is integrated to projective representations ofSU(2,2). These representations are shown to be equivalent to the representations of the interpolated discrete series ofSU(2,2). Using this result we give a formula for the two-point Wightman function for arbitrary spin and dimension of the field. Finally we discuss the limit when the dimension tends to the canonical value.     

Polynomial poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Hidden quantum groups inside Kac-Moody algebra

A lattice analogue of the Kac-Moody algebra is constructed. It is shown that the generators of the quantum algebra with the deformation parameterq=exp(iπ/k+h) can be constructed in terms of generators of the lattice Kac-Moody algebra (LKM) with the central chargek. It appears that there exists a natural correspondence between representations of the LKM algebra and the finite dimensional quantum group. The tensor product for representations of the LKM algebra and the finite dimensional quantum algebra is suggested.     

Extremal vectors for Verma-type representations of su(2, 2)

Starting from the Verma modules of the algebra sl(4, ?) we explicitly construct factor representations of the algebra su(2, 2) which are connected with unitary representation of group SU(2, 2). We find a full set of extremal vectors for this kind of representations, so we can solve explicitly the problem of irreducibility of these representations.     

Cyclic monodromy matrices forsl(n) trigonometricR-matrices

The algebra of monodromy matrices forsl(n) trigonometricR-matrix is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product ofL-operators. Cocommutativity of representations is discussed and intertwiners for factorizable representations are written through the Boltzmann weights of thesl(n) chiral Potts model.     

Irreducible representations of Virasoro-toroidal Lie algebras

Toroidal Lie algebras and their vertex operator representations were introduced in [MEY] and a class of indecomposable modules were investigated. In this work, we extend the toroidal algebra by the Virasoro algebra thus constructing a semi-direct product algebra containing the toroidal algebra as an ideal and the Virasoro algebra as a subalgebra. With the use of vertex operators and certain oscillator representations of the Virasoro algebra it is proved that the corresponding Fock space gives rise to a class of irreducible modules for the Virasoro-toroidal algebra.To A. John Coleman on the occasion of his 75th birthday     

Sugawara construction for higher genus Riemann surfaces

By the classical genus zero Sugawara construction one obtains representations of the Virasoro algebra from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type). In this lecture, the classical construction is recalled first. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in a joint work with O. K. Sheinman. We were able to show that also in the higher genus, multi-point situation one obtains (from representations of the global algebras of affine type) representations of a centrally extended algebra of meromorphic vector fields on Riemann surfaces. The latter algebra is a generalization of the Virasoro algebra to higher genus.     

Hilbert space representations of Lie algebras

We describe a new approach to the general theory of unitary representations of Lie groups which makes use of the Gelfand-Segal construction directly on the universal enveloping algebra of any Lie algebra. The crucial observation is that Nelson's theory of analytic vectors allows the characterisation of certain states on the universal enveloping algebra such that the corresponding representations of the universal enveloping algebra are the infinitesimal part of unitary representations of the associated simply connected Lie group. In the first section of the paper we show that with the aid of Choquet's theory of representing measures one can derive a simple new approach to integral decomposition theory along these lines.In the second section of the paper we use these methods to study the irreducible unitary representations of general semi-simple Lie groups. We give a simple proof that theK-finite vectors studied by Harish-Chandra [5] are all analytic vectors. We also give new proofs of some of Godement's results [2] characterising spherical functions of height one, at least for unitary representations. Compared with [2] our method has the possible advantage of obtaining the characterisations by infinitesimal methods instead of using an indirect argument involving functions on the group. We point out that while being purely algebraic in nature, this approach makes almost no use of the deep and difficult theorems of Harish-Chandra concerning the universal enveloping algebra [5].Our work is done in very much the same spirit as that of Power's recent paper [8]. The main difference is that by concentrating on a more special class of positive states we are able to carry the analysis very much further without difficulty.     

Solutions of the Yang-Baxter Equation and Temperley-Lieb Algebraic Structure

The necessary and sufficient condition for the representations of Temperley-Lieb algebra arising from representations of Heck algebra is presented. Using the condition, the representations of braid group with a quadratic reduction relation are divided into two kinds. One has a representation of Temperley-Lieb algebra behind it and another has not. Some concrete examples are discussed with the condition and their Baxterizations are also mentioned.     

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest ₃ ×SO(8) structure in this framework.     

Vertex representations forN-toroidal Lie algebras and a generalization of the Virasoro algebra

Vertex representations are obtained for toroidal Lie algebras for any number of variables. These representations afford representations of certainn-variable generalizations of the Virasoro algebra that are abelian extensions of the Lie algebra of vector fields on a torus.Work supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Dynamical symmetries and conservation laws for Korteweg-de Vries equation

The KdV-equation in two space time dimensions with the set of rapidly decreasing test functions as initial conditions is treated in the setting of nonlinear group and Lie algebra representations. The topological properties of the direct and inverse scattering mappings are discussed in detail.The algebra of continuous constants of motion turns out to be generated as in the linear case by three constants of motion and an extension of a representation of the e₂ Lie algebra on space-time symmetries to its enveloping algebra. The integrability of these representations is studied.It is further proved that the “moment problem” does not have a unique solution in this setting.The existence of noncommutative algebras of smooth time independent constants of motion is pointed out.     

Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras

The homology of the Lie algebra of algebraic vector fields in the complex line with trivial 3-jet at the point 0 with the coefficients in irreducible highest weight representations of the Virasoro Lie algebra is calculated. The same is done for vector fields with trivial 1-jets at two distinguished points. The class of quasi- finite representations of the Virasoro Lie algebra naturally arises which is the substitute for the class of finite-dimensional representations. The similar results for Kac-Moody Lie algebras are given as well as some conjectures and announcements.     

Unitary representations of the Virasoro and super-Virasoro algebras

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.     

Realisations of the representations of para-Fermi algebras—Part IV: Fock spaces of para-Bose and para-Fermi operators

The representations of the para-Fermi algebra in the Fock spaces of para-Bose and para-Fermi operators are constructed. The unitary equivalence of different representations is proved. The Bardeen-Cooper-Schrieffer pair creation and annihilation operators and the four fermion interaction appear as particular realisations of the para-Fermi algebra. The para-Fermi algebra representations in quantum mechanics are discussed.     

Representations of BRS algebra

In the canonical formulation of gauge theories the BRS transformation plays a fundamental rôle. The generator of this transformation along with the ghost number forms an algebra called the BRS algebra. Certain properties of this algebra are essential to the proof of unitarity of the S matrix in the physical sector and also to the discussion of color confinement.In the present paper we present all the possible representations of the BRS algebra in the light of indefinite metric.