Unitary representations of some infinite-dimensional Lie algebras motivated by string theory on AdS3

We consider some unitary representations of infinite-dimensional Lie algebras motivated by string theory on AdS₃. These include examples of two kinds: the A,D,E type affine Lie algebras and the N=4 superconformal algebra. The first presents a new construction for free field representations of affine Lie algebras. The second is of a particular physical interest because it provides some hints that a hybrid of the NSR and GS formulations for string theory on AdS₃ exists.     

Graded manifolds,supermanifolds and infinite-dimensional Grassmann algebras

A new approach to superdifferentiable functions of Grassmann variables is developed, which avoids ambiguities in odd derivatives. This is used to give an improved definition of supermanifold over a finite-dimensional Grassmann algebra. A natural embedding of super-manifolds over Grassmann algebras with increasing number (L) of generators is developed, and thus a limit asL tends to infinity is possible. A correspondence between graded manifolds and supermanifolds is constructed, extending results of [5] and [8].Research supported by the SERC under advanced research fellowship number B/AF/687     

Symplectic reduction,BRS cohomology,and infinite-dimensional Clifford algebras

This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical finite dimensional BRS procedure and relate it to Marsden-Weinstein reduction. This leads to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. We then discuss infinite-dimensional Clifford algebras and their spin representations. We find that in the infinite-dimensional case, the analog of the finite-dimensional construction of Lie algebra cohomology breaks down, the obstruction (anomaly) being the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation of opposite class kills the obstruction and gives rise to a cohomology theory and a quantization procedure. We discuss the gradings and Hermitian structures on the absolute and relative complexes.     

Usage of infinite-dimensional nuclear algebras in superanalysis

An infinite-dimensional topological algebra is defined as an inductive limit of finite-dimensional -commutative Banach algebras. This algebra has some desirable properties for the algebra of supernumbers, on which we can develop a satisfactory theory of superanalysis.     

Coherent states and infinite-dimensional lie algebras: An outlook

Summary The possible extension of the notion of generalized coherent state to the case of infinite-dimesional affine Lie algebras is discussed with special attention to the resulting topological structure of the coherent states manifold, and to its connection with the structure of the algebra. The relevance for the solution of nonlinear dynamical systems equations of motion is briefly reviewed. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

Boson representations of symplectic algebras

A reduction of the boson representation of the algebra of the noncompact groupSp(4k, R), k>0, to its subgroupSU(k) is realized. The reduction scheme has two main branches: one through the totally symmetric unitary representations of the maximal compact subalgebra u(2k); the other through the ladder representations of the noncompact subalgebrau(k, k). Both reductions are accomplished by means of the same set of Hermitian operators, but taken in different order. The case ofk=3, for the groupSp(12,R), used in the interacting vector boson model, is discussed in more detail.     

On representations of Hecke algebras

In this report we review some facts about representation theory of Hecke algebras. For Hecke algebras we adapt the approach of A. Okounkov and A. Vershik [Selecta Math., New Ser., 2 (1996) 581], which was developed for the representation theory of symmetric groups. We justify explicit construction of idempotents for Hecke algebras in terms of Jucys-Murphy elements. Ocneanu's traces for these idempotents (which can be interpreted as q-dimensions of corresponding irreducible representations of quantum linear groups) are presented. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. This work was supported in part by the grants INTAS 03-51-3350 and RFBR 05-01-01086-a.     

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 $n=4$.     

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.     

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.     

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.     

Fock representations of ZF algebras and R-matrices

Letters in Mathematical Physics - A variation of the Zamolodchikov–Faddeev algebra over a finite-dimensional Hilbert space $${mathcal {H}}$$ and an involutive unitary R-Matrix S is studied....     

Induced representations of quantum kinematical algebras

We construct the induced representations of the null-plane quantum Poincaré and quantum kappa Galilei algebras in (1+1) dimensions. The induction procedure makes use of the concept of module and is based on the existence of a pair of Hopf algebras with a nondegenerate pairing and dual bases.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

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     

Regular * representations of lie algebras

We prove the existence of a * product on the cotangent bundle of a parallelizable manifold M. When M is a Lie group the properties of this * product allow us to define a linear representation of the Lie algebra of this group on L ²(G), which is, in fact, the one corresponding to the usual regular representation of G.Chargé de recherches au FNRS.     

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.     

Noncommutative differential geometry, and the matrix representations of generalised algebras

The underly ing algebra I or a noncummutative geometry is taken to be a matrix algebra, and the set of derivatives the ad joint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of 1-firms is at free module over the algebra of matrices. The concept of a generalised algebra is delined and it is shown that this is required in order for the space of 2-forms to exist, The exterior derivative is generalised for higher-order forms and these are also shown to he free modules over the matrix algebra. Examples of mappings that preserve the differential Structure are peen, Also giken are four examples of matrix generalised algebras, and the corresponding noncommutntive geometries, including the cases where the generalised algebra corresponds to a representation of a Lie algebra or a q-deformed algebra.