One-Parameter Indecomposable Representation of SPL(2,1) Superalgebra

Using inhomogeneous boson–fermion realization one-parameter indecomposable representation of the SPL(2,1) superalgebra is studied on subspace and quotient spaces of the universal enveloping algebra of Heisenberg-Weyl superalgebra. PACS: 12.60.Jv; 03.65.FdThe author was supported financially by Shenzhen Institute of Information Technology.     

SPL(2，1)超代数的不可分解表示和不可约表示

利用ＳＰＬ（２,１）的非齐次玻色 费米实现,研究了ＳＰＬ（２,１）在Ｈｅｉｓｅｎｂｅｒｇ Ｗｅｙｌ超代数的广义包络代数的子空间和商空间上的不可分解表示和不可约表示,并给出了它的全部有限维不可约表示 关键词：     

Contraction of graded su(2) algebra

The Inönu-Wigner contraction scheme is extended to Lie superalgebras. The structure and representations of extended BRS algebra are obtained from contraction of the graded su(2) algebra. From cohomological consideration, we demonstrate that the graded su(2) algebra is the only superalgebra which, on contraction, yields the full BRS algebra.     

u q(2,1) noncompact quantum algebra: Intermediate discrete series of unitary irreducible representations

The unitary irreducible representations of the u _q(2,1) quantum algebra that belong to the intermediate discrete series are considered. The q analog of the Mickelsson-Zhelobenko algebra is developed. Use is made of the U basis corresponding to the reduction u _q(2,1) ? u _q(2). Explicit formulas for the matrix elements of the generators are obtained in this basis. The projection operator that projects an arbitrary vector onto the extremal vector of the intermediate-series representation is found.     

The representations and coupling coefficients of su(n); application to su(4)

Analytical expressions for the matrices and an explicit algorithm for computing Clebsch-Gordan coupling coefficients are given forsu(4) in au(3)-coupled basis as an example of the construction for anysu(n) in au(n−1) basis. The results areinduced from the known results foru(3) by means of the vector-coherent-state (VCS) theory of induced representations. The important recent result that makes this possible is the discovery that a complete set of shift tensors for the finitedimensional representations of reductive Lie algebras can be induced, by VCS methods, from those of suitably defined subalgebras.     

Indecomposable representations with invariant inner product

Consequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space \(\mathfrak{H}\) are studied. If π has an irreducible subrepresentation π₁ on a subspace \(\mathfrak{H}_1 \) , it is shown that there exists an invariant subspace \(\mathfrak{H}_2 \) of \(\mathfrak{H}\) containing \(\mathfrak{H}_1 \) and satisfying the following conditions: (1) the representation π ₁ ^# =π mod \(\mathfrak{H}_2 \) on \(\mathfrak{H}\) mod \(\mathfrak{H}_2 \) is conjugate to the representation (π₁, \(\mathfrak{H}_1 \) ), (2) \(\mathfrak{H}_1 \) is a null space for the inner product, and (3) the induced inner product on \(\mathfrak{H}_2 \) mod \(\mathfrak{H}_1 \) is non-degenerate and invariant for the representation $$\pi _2 = (\pi _2 |_{\mathfrak{H}_2 } )\bmod \mathfrak{H}_1 ,$$ a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with \(\mathfrak{H}_1 \) =space of longitudinal photons and \(\mathfrak{H}_2 \) =the space defined by the subsidiary condition.     

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.     

Finite dimensional representations of theSU(2)-current Lie algebra

We give a complete classification of the finite dimensional solutions for the Lie functional equations ofSU(2).     

Fock representations and BRST cohomology inSL(2) current algebra

We investigate the structure of the Fock modules overA ₁ ⁽¹⁾ introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.     

The su q(2) algebra in the off-diagonal basis and applications to quantum optics

We consider a new exactly solvable nonlinear quantum model as a Hamiltonian defined in terms of the generators of the su _q(2) algebra. The corresponding matrix elements of finite rotations (the q-deformed Wigner d functions) are introduced. It is shown that the quantum optical model of the three-wave interaction has an approximate su _q(2) dynamical symmetry given by this Hamiltonian. Such q symmetry allows us to investigate the spectral and dynamical properties of the three wave model through new perturbation techniques.     

Vertex operator representations of the quantum affine algebra U q(B r (1) )

We construct the level one vertex operator representations of the q-deformation U _q(B _r ⁽¹⁾ ) of the affine Kac-Moody algebra B _r ⁽¹⁾ . Beside the q-deformed vertex operators introduced by Frenkel and Jing, this construction involves a q-deformation of free fermionic fields.     

Models of Q algebra representations

This article gives a review of various straightforward models ofQ algebra representations. This is done using one and two variable function space models of theq-analogues of Lie enveloping algebras. The algebras considered are the quantum algebraU _q (su ₂ ) and aq analogue of the oscillator algebra. We present only the general framework and refer the reader to references of the joint work of the author and Willard Miller, Jr.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

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.