Remark on the integrability of some representations of the semisimple Lie algebras

It is discussed how boundedness of the quadratic Casimir operator in skew-symmetric representation τ of semisimple Lie algebra can simplify the proof of integrability of τ.     

Unitary representations of basic classical Lie superalgebras

We have obtained all the finite-dimensional unitary irreps of gl(m|n) and C(n), which also exhaust such irreps of all the basic classical Lie superalgebras. The lowest weights of such irreps are worked out explicitly. It is also shown that the contravariant and covariant tensor irreps of gl(m|n) are unitary irreps of type (1) and type (2) respectively, explaining the applicability of the Young diagram method to these two types of tensor irreps.     

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.     

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     

Leibniz algebras associated with representations of filiform Lie algebras

In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra $n_{n,1}$. We introduce a Fock module for the algebra $n_{n,1}$ and provide classification of Leibniz algebras $L$ whose corresponding Lie algebra $L/I$ is the algebra $n_{n,1}$ with condition that the ideal $I$ is a Fock $n_{n,1}$-module, where $I$ is the ideal generated by squares of elements from $L$.We also consider Leibniz algebras with corresponding Lie algebra $n_{n,1}$ and such that the action $I\times n_{n,1}\to I$ gives rise to a minimal faithful representation of $n_{n,1}$. The classification up to isomorphism of such Leibniz algebras is given for the case of $n=4$.     

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.     

On the physical representations of an infinite-dimensional Lie algebra

Irreducible Hermitian representations of the infinite-dimensional Lie algebra, presented by J. Formánek [Czech. J. Phys.B16 (1966), 1,281] and suitable for the classification of elementary particles, are characterized by assumptions I–III. All representations of this kind are found explicitly.     

Lie algebras and representations of continuous groups

It is shown that every finitely generated continuous group has a subgroup generated by its infinitesimal transformations. This subgroup has a group algebra which is the Lie algebra of the group. By obtaining complete systems in the Lie algebra and complete rectangular arrays, it is shown that these can yield matrix representations of the continuous group. Illustrative examples are given for the rotation groups and for the full linear groups. It would seem that all the finite motion representations can be obtained by these methods, including spin representations of rotation groups. But the completeness of the method is not here demonstrated.     

Parabosons,parafermions, and explicit representations of infinite-dimensional algebras

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(∞) and of the Lie superalgebra osp(1|∞). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labeled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. Alternatively, the basis vectors can be expressed as semi-standard Young tableaux.     

Unitary representations of some infinite dimensional groups

We construct projective unitary representations of (a) Map(S ¹;G), the group of smooth maps from the circle into a compact Lie groupG, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of Map(S ¹;T), whereT is a maximal torus ofG, can be extended to representations of Map(S ¹;G),     

Quasi-Hamiltonian mechanics associated to representations of Lie algebras

For relative classical mechanics we construct a quasi-hamiltonian formalism associated to special representations of Lie algebras. This formalism is natural. For the case of the adjoint representation, this construction reduces to the usual absolute hamiltonian formalism on the dual spaces to Lie algebras.     

The tensor product of one-particle representations of an infinite-dimensional Lie algebra

The recently proposed infinite-dimensional Lie algebra as a model of a symmetry scheme is studied from the point of view of its representations. We construct the tensor product of two one-particle representations of this algebra and study the reduction problem. A new series of representations having non-linear mass formulas is found. Some physical consequences for two-particle states are also discussed.The authors wish to thank Prof. V. Votruba for constant interest shown throughout this work and J. Formánek for valuable discussions.     

Long string dynamics in pure gravity on AdS3

Journal of Experimental and Theoretical Physics - We study the classical dynamics of a completion of pure AdS3 gravity, whose only degrees of freedom are boundary gravitons and long strings. We...     

Representations of Lie algebras from representations of quantum groups

In paper [*] (P. Moylan: Czech. J. Phys., Vol. 47 (1997), p. 1251) we gave an explicit embedding of the three dimensional Euclidean algebra (2) into a quantum structure associated with U _q(so(2, 1)). We used this embedding to construct skew symmetric representations of (2) out of skew symmetric representations of U _q(so(2, 1)). Here we consider generalizations of the results in [*] to a more complicated quantum group, which is of importance to physics. We consider U _q(so(3, 2)), and we show that, for a particular representation, namely the Rac representation, many of the results in [*] carry over to this case. In particular, we construct representations of so(3, 2), P(2, 2), the Poincaré algebra in 2+2 dimensions, and the Poincaré algebra out of the Rac representation of U _q(so(3, 2)). These results may be of interest to those working on exploiting representations of U _q(so(3, 2)), like the Rac, as an example of kinematical confinement for particle constituents such as the quarks.     

On the integrability of representations of finite dimensional real Lie algebras

We give an integrability criterion for Lie algebra representations in a reflexive Banach space. Applications are given to skewsymmetric Lie algebra representations in Hilbert spaces and to essential skewadjointness of a sum of two skewadjoint operators.     

Analytic vectors and irreducible representations of nilpotent Lie groups and algebras

Let U be a unitary irreducible locally faithful representation of a nilpotent Lie group G, U the universal enveloping algebra of G, M a simple module on U with kernel Ker dU, then there exists an automorphism of U keeping ker dU invariant such that, after transport of structure, M is isomorphic to a submodule of the space of analytic vectors for U.     

New integrable hierarchies from vertex operator representations of polynomial Lie algebras

We give a representation–theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupled to each Drinfeld–Sokolov of A, B, C, D hierarchies and to construct their soliton solutions.     

Twist of Lie algebras by a rank-3 subalgebra

A new nonstandard deformation of all types of classical Lie algebras is constructed by means of twisting based on a six-dimensional subalgebra. This is an extension of extended twists introduced by Kulish et al. It is also shown that the new nonstandard so(3, 2) has a close connection with the symmetry of a discrete analog of the Klein-Gordon equation in (1+2) spacetime.