Possible representations of local current algebra

Mass operators which lead to possible representations of local (vector) current algebra are discussed.     

Finite-dimensional representations of a shock algebra

Abstruct The algebra describing a shock measure in the asymmetric simple exclusion model, seen from a second class particle, has finite-dimensional representations if and only if the asymmetry parameterp of the model and the left and right asymptotic densitiesp _± of the shock satisfy [(1−p)/p] ^r =p ₋(1−p ₊)/p ₊(1−p ₋) for some integerr≥1; the minimal dimension of the representation is then 2r. These representations can be used to calculate correlation functions in the model.     

Instantons and representations of an associative algebra

We give the correspondence between instantons onS ⁴ and some representations of an associative algebra. For the given structure group, we get simultaneous imbeddings to (the inductive limit) of the moduli spaces for instantons onS ⁴ of all instanton numbers.     

Oscillator representations of the 2D-conformal algebra

We display irreducible representations of the Virasoro algebra (group of diffeomorphisms of the circle) for any value of the central chargec (central extension defined by a cocycle) and of the highest weight, where the Ka determinants do not vanish. The construction is done in terms of a simple bosonic free field. The unitarity of the representation is discussed, and it is realized with non-trivial hermiticity properties of the free field if<(c-1)/24. In the particular case of the central charge (c=1/2) corresponding to the Ising model, the three unitary irreducible representations (=0, 1/16, 1/2) are realized in terms of the anticommuting oscillators of the free fields of the Neveu-Schwarz-Ramond model.     

On the representations of quantum oscillator algebra

It is shown that for q<1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SU _q (1,1) and E(2) in Schwinger's way.     

On non-local representations of the ageing algebra

The ageing algebra is a local dynamical symmetry of many ageing systems, far from equilibrium, and with a dynamical exponent z=2$z=2$. Here, new representations for an integer dynamical exponent z=n$z=n$ are constructed, which act non-locally on the physical scaling operators. The new mathematical mechanism which makes the infinitesimal generators of the ageing algebra dynamical symmetries, is explicitly discussed for an n-dependent family of linear equations of motion for the order-parameter. Finite transformations are derived through the exponentiation of the infinitesimal generators and it is proposed to interpret them in terms of the transformation of distributions of spatio-temporal coordinates. The two-point functions which transform co-variantly under the new representations are computed, which quite distinct forms for n even and n odd. Depending on the sign of the dimensionful mass parameter, the two-point scaling functions either decay monotonously or in an oscillatory way towards zero.     

Poisson structures due to Lie algebra representations

Using a unitary solution of the classical Yang-Baxter equation on a Lie algebraG we describe a particular way of constructing homogeneous quadratic Poisson structures on the dual of aG-moduleV and study some local features of the symplectic foliation due to the involutive distribution of the Hamiltonian vector fields. We also give some examples where the symplectic leaves are explicitly calculated.     

Integrable models through representations of the hecke algebra

We present new solutions to the Yang-Baxter equation through representations of the Hecke algebra. The generators of the Hecke algebra are considered as Boltzmann weights for face models. The heights live in a graph. These models were conjectured to be integrable by Di Francesco and Zuber. We prove integrability for some of the suggested models by building explicitly the Boltzmann weights. Some relations that the Boltzmann weights satisfy and the consequence on partition functions with various boundary conditions are also discussed.     

Induced representations of the two-parameter Jordanian quantum algebra

We construct induced infinite-dimensional representations of the two-parameter quantum algebraUg,h(gl(2)) which is in duality with the deformationGLg,h(2). The representations depend on two representation parameters, but split into one-parameter representations of a one-generator central subalgebra and the three-generator Jordanian quantum subalgebraU (sl(2)), =g + h. The representations of the latter can be mapped to representations in one complex variable, which give anew deformation of the standard one-parameter vector-field realization ofsl(2). These infinite-dimensional representations are reducible for some values of the representation parameters, and then we obtain canonically the finite-dimensional representations ofU (sl(2)). Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Permanent address of V.K.D.     

Two-rowed Hecke algebra representations at roots of unity

In this paper, we initiate a study into the explicit construction of irreducible representations of the Hecke algebraH _n (q) of typeA _n-1 in the non-generic case whereq is a root of unity. The approach is via the Specht modules ofH _n (q) which are irreducible in the generic case, and possess a natural basis indexed by Young tableaux. The general framework in which the irreducible non-genericH _n (q)-modules are to be constructed is set up and, in particular, the full set of modules corresponding to two-part partitions is described. Plentiful examples are given.Presented at the 4th International Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

About one class of representations of the Lie algebra

LetR(G) be the skewsymmetric representation of the algebraG characterized by the following main property: ifGG is some subalgebra ofG (possible noncompact) thenR(G) is integrable and reducible in the direct sum of irreducible representations of subalgebraG.The paper is devoted to the development of the elementary theory of the described representations, culminating in the proof of one version of Schur's lemma.     

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.     

The unitary irreducible representations of the quantum heisenberg algebra

The goal of this work is to describe the irreducible representations of the quantum Heisenberg algebra and the unitary irreducible representation of one of its real forms. The solution of this problem is obtained through the investigation of theleft spectrum of the quantum Heisenberg algebra using the result about spectra of generic algebras of skew differential operators (cf. [R]).     

Unitary equivalence of Fock representations on the Weyl algebra

A necessary and sufficient condition for unitary equivalence of pure quasifree states over the Weyl algebra is proved. Some partial results on states over the Weyl algebra are formulated in Theorem 1, and Lemmas 1, 4, 5 and 6.     

On the skew representations of the quantum affine algebra

An explicit realization of the skew representations of the quantum affine algebra U _q (gl _n ) is given. It is used to identify these representations in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand-Tsetlin character (orq-character).     

Indecomposable representations of the algebra su(2,1)

Representations of the Lie algebra sl(3) with highest weight are analyzed. Invariant subspaces of indecomposable representations are determined. We study the decomposition of these representations with respect to the subalgebras su(2) and su(1,1) (in their obvious imbedding in su(2,1)).For special cases this decomposition gives indecomposable non multiplicity free representations (indecomposable pairs) with highest weight. These were discussed in [1] and appear also in the decomposition so(3,2) su(1,1) of the Rac representation, [7].     

多项式角动量代数的代数表示及实现

本文利用三参数李群求代数表示的方法求出多项式角动量代数的代数表示及其酉表示，找到一个能同时承载李代数及相对应的多项式角动量代数的基底，并在该基底下求出两种代数之间的联系，利用该联系则也可求出多项式角动量代数的代数表示.最后求出多项式角动量代数的单玻色实现及其在有限维多项式函数空间的微分实现. 关键词： 多项式角动量代数 Higgs代数 su(2)代数