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The exotic bialgebra S03, defined by a solution of the Yang-Baxter equation, which is not a deformation of the trivial, is considered. Its FRT dual algebra $s03_F$ is studied. The Baxterisation of the dual algebra is given in two different parametrisations. The finite-dimensional representations of $s03_F$ are considered. Diagonalisations of the braid matrices are used to yield remarkable insights concerning representations of the L-algebra and to formulate the fusion of finite-dimensional representations. Possible applications are considered, in particular, an exotic eight-vertex model and an integrable spin-chain model., Comment: 24 pages, Latex; V2: revised subsection 4.1, added 9 references, to appear in Annales Henri Poincare in the volume dedicated to D. Arnaudon

The aim of this paper is to apply systematically to AdS_4 some modern tools in the representation theory of Lie algebras which are easily generalised to the supersymmetric and quantum group settings and necessary for applications to string theory and integrable models. Here we introduce the necessary representations of the AdS_4 algebra and group. We give explicitly all singular (null) vectors of the reducible AdS_4 Verma modules. These are used to obtain the AdS_4 invariant differential operators. Using this we display a new structure - a diagram involving four partially equivalent reducible representations one of which contains all finite-dimensional irreps of the AdS_4 algebra. We study in more detail the cases involving UIRs, in particular, the Di and the Rac singletons, and the massless UIRs. In the massless case we discover the structure of sets of 2s_0-1 conserved currents for each spin s_0 UIR, s_0=1,3/2,... All massless cases are contained in a one-parameter subfamily of the quartet diagrams mentioned above, the parameter being the spin s_0. Further we give the classification of the so(5,C) irreps presented in a diagramatic way which makes easy the derivation of all character formulae. The paper concludes with a speculation on the possible applications of the character formulae to integrable models., Comment: 30 pages, 4 figures, TEX-harvmac with input files: amssym.def, amssym.tex, epsf.tex; version 2 1 reference added; v3: minor corrections; v.4: minor corrections, v.5: minor corrections to conform with version in J. Phys. A: Math. Gen; v.6.: small correction and addition in subsections 4.1 & 4.2

In the classification of solutions of the Yang--Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding dual algebras. We then study the representations of the latter. We are also interested in the Baxterisation of these $R$-matrices and in the corresponding quantum planes., Comment: 11 pages, LaTeX2e, requires ws-procs9x6.cls; using packages: amsfonts, amsmath, amssymb; talk by D.A. at 5-th International Workshop "Lie Theory and Its Applications in Physics", Varna, Bulgaria, (June 2003), to appear in the Proceedings (World Scientific, Singapore, 2004)

We give character formulae for the positive energy unitary irreducible representations of the N-extended D=4 conformal superalgebras su(2,2/N). Using these we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4/N). Our derivations use results from the representation theory of su(2,2/N) developed already in the 80s., Comment: 81 pages, input files: harvmac, amssym.def, amssym.tex; corrected presentation on odd reflections

We give new solutions of the quantum conformal deformations of the full Maxwell equations in terms of deformations of the plane wave. We study the compatibility of these solutions with the conservation of the current. We also start the study of quantum linear conformal (Weyl) gravity by writing the corresponding q-deformed equations., Comment: 18 pages, Latex2e; v2: corrected misprints; v3: small text improvement; v4: misprints in f-la (3.14) corrected

We consider the unitarity of D=9,10,11 conformal supersymmetry using the recently established classification of the UIRs of the superalgebras osp(1|2n,R)., Comment: 7 pages, LATEX2e{czjphys} (amsmath,amssymb); v3: important corrections, Plenary talk by VKD at the XIII Colloquium "Quantum Groups and Integrable Systems", Prague, 17-19.6.2004; to appear in Czech. J. Phys.; v4: small corrections to coincide with journal version

We give the classification of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R)., Comment: 20 pages, LATEX2e (revtex4,amsmath,amssymb), Plenary talk by VKD at X International Conference on Symmetry Methods in Physics, Yerevan, 13-21.8.2003; added acknowledgements; corrected misprints

The present paper contains two interrelated developments. First, are proposed new generalized Verma modules. They are called k-Verma modules, k\in N, and coincide with the usual Verma modules for k=1. As a vector space a k-Verma module is isomorphic to the symmetric tensor product of k copies of the universal enveloping algebra U(g^-), where g^- is the subalgebra of lowering generators in the standard triangular decomposition of a simple Lie algebra g = g^+ \oplus h \oplus g^- . The second development is the proposal of a procedure for the construction of multilinear intertwining differential operators for semisimple Lie groups G . This procedure uses k-Verma modules and coincides for k=1 with a procedure for the construction of linear intertwining differential operators. For all k central role is played by the singular vectors of the k-Verma modules. Explicit formulae for series of such singular vectors are given. Using these are given explicitly many new examples of multilinear intertwining differential operators. In particular, for G = SL(2,R) are given explicitly all bilinear intertwining differential operators. Using the latter, as an application are constructed (n/2)-differentials for all n\in 2N, the ordinary Schwarzian being the case n=4. As another application, in a Note Added we propose a new hierarchy of nonlinear equations, the lowest member being the KdV equation., Comment: 30 pages, plain TEX with harvmac; corrected misprints

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev-Reshetikhin-Takhtajan RTT prescription with non-triangular R-matrices which are denoted $R_{03}$ and $R_{14}$ in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1/1). We give the spectral decomposition which involves two, resp., three, projectors. These projectors are then used to provide the Baxterisation procedure with one, resp., two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess-Zumino prescription. In all these constructions there appear non-standard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators., Comment: 14 pages, LaTeX2e, using packages: cite, amsfonts, amsmath, subeqn; v2: small improvements, coincides with journal version

A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding quantisation method. After a short discussion this method is translated step-by-step to a framework based on difference operators. To restrict the resulting plethora of possible quantisations additional assumptions motivated by simplicity and plausibility are required. Multiplicative difference operators and the corresponding q-Borel kinematics are given on the circle and its N-point discretisation; the connection to q-deformations of the Witt algebra is discussed. For a "natural" choice of the q-kinematics a corresponding q-difference evolution equation is obtained. This study shows general difficulties for a generalisation of a physical theory from a known one to a "new" framework., Comment: 19 pages, LaTeX, using packages: pictex, amssymb; v.2: small corrections, to appear in Rep. Math. Phys

We use the group-theoretic interpretation of the AdS/CFT correspondence which we proposed earlier in order to lift intertwining operators acting between boundary conformal representations to intertwining operators acting between bulk conformal representations. Further, we present the classification of the positive energy (lowest weight) unitary irreducible representations of the D=6 superconformal algebras $osp(8^*/2N)$., Comment: 22 pages, LaTeX2e; Lectures at the 1st Summer School in Modern Mathematical Physics, Sokobanja, Yugoslavia, 13-25.8.2001; to appear in the Proceedings of the School

We find the exotic matrix bialgebras which correspond to the two non-triangular nonsingular 4x4 R-matrices in the classification of Hietarinta, namely, R_{S0,3} and R_{S1,4}. We find two new exotic bialgebras S03 and S14 which are not deformations of the of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a special case of R_{S1,4} in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GL_{p,q}(2)., Comment: 33 pages, LaTeX2e, using packages: cite,amsfonts,amsmath,subeqn; reference updated; v3: corrections in subsection 3.4

We give a constructive classification of the positive energy (lowest weight) unitary irreducible representations of the D=6 superconformal algebras osp(8*/2N). Our results confirm all but one of the conjectures of Minwalla (for N=1,2) on this classification. Our main tool is the explicit construction of the norms of the states that has to be checked for positivity. We give also the reduction of the exceptional UIRs., Comment: 27 pages, TeX with harvmac, amssym.def, amssym.tex; v.2: minor corrections and references added; v.3: minor corrections; v.4: to appear in J. Phys. A

A deformation of the harmonic oscillator algebra associated with the Morse potential and the SU(2) algebra is derived using the quantum analogue of the anharmonic oscillator. We use the quantum oscillator algebra or $q$-boson algebra which is a generalisation of the Heisenberg-Weyl algebra obtained by introducing a deformation parameter $q$. Further, we present a new algebraic realization of the $q$-bosons, for the case of $q$ being a root of unity, which corresponds to a periodic structure described by a finite-dimensional representation. We show that this structure represents the symmetry of a linear lattice with periodic boundary conditions., Comment: LATEX2e, 10 pages, v2: few misprints corrected, added Journal-ref

In the classification of Hietarinta, three triangular $4\times 4$ $R$-matrices lead, via the FRT formalism, to matrix bialgebras which are not deformations of the trivial one. In this paper, we find the bialgebras which are in duality with these three exotic matrix bialgebras. We note that the $L-T$ duality of FRT is not sufficient for the construction of the bialgebras in duality. We find also the quantum planes corresponding to these bialgebras both by the Wess-Zumino R-matrix method and by Manin's method., Comment: 25 pages, LaTeX2e, using packages: cite, amsfonts, amsmath, subeqn

Combined $(q,h)$-deformations proposed by Kupershmidt and Ballesteros-Herranz-Parashar are studied. In each case a transformation is shown to lead to an equivalent, standard $q$-deformation. We briefly indicate that appropriate singular limits of the same type of transformations can however lead from standard biparametric $(p,q)$-deformations to non-hybrid but biparametric nonstandard $(g,h)$ ones. Finally a case of hybrid $(q,h)$-deformation is recalled, related to the superalgebra $GL(1|1)$., Comment: 17 pages, LaTeX2e, using packages: amsfonts, amsmath; V2: added preprint number, corrected misprints; J. Math. Phys., to appear

L\'evy processes on bialgebras are families of infinitely divisible representations. We classify the generators of L\'evy processes on the compact forms of the quantum algebras $U_q(g)$, where $g$ is a simple Lie algebra. Then we show how the processes themselves can be reconstructed from their generators and study several classical stochastic processes that can be associated to these processes., Comment: 13 pages, LATEX file, ASI-TPA/13/99 (TU Clausthal); 6/99 (Preprint-Reihe Mathmatik, Univ. Greifswald)

We give a group-theoretic interpretation of the AdS/CFT correspondence as relation of representation equivalence between representations of the conformal group describing the bulk AdS fields $\phi$ and the coupled boundary fields $\phi_0$ and ${\cal O}$. We use two kinds of equivalences. The first kind is equivalence between bulk fields and boundary fields and is established here. The second kind is the equivalence between coupled boundary fields. Operators realizing the first kind of equivalence for special cases were given by Witten and others - here they are constructed in a more general setting from the requirement that they are intertwining operators. The intertwining operators realizing the second kind of equivalence are provided by the standard conformal two-point functions. Using both equivalences we find that the bulk field has in fact two boundary fields, namely, the coupled boundary fields. Thus, from the viewpoint of the bulk-boundary correspondence the coupled fields are on an equal footing. Our setting is more general since our bulk fields are described by representations of the Euclidean conformal group $G=SO(d+1,1)$, induced from representations $\tau$ of the maximal compact subgroup $SO(d+1)$ of $G$. From these large reducible representations we can single out representations which are equivalent to conformal boundary representations labelled by the conformal weight and by arbitrary representations $\mu$ of the Euclidean Lorentz group $M=SO(d)$, such that $\mu$ is contained in the restriction of $\tau$ to $M$. Thus, our boundary-to-bulk operators can be compared with those in the literature only when for a fixed $\mu$ we consider a 'minimal' representation $\tau=\tau(\mu)$ containing $\mu$., Comment: 25 pages, TEX file using harvmac.tex; v2: misprints corrected; to appear in Nuclear Physics B

We construct finite-dimensional irreducible representations of two quantum algebras related to the generalized Lie algebra $\ssll (2)_q$ introduced by Lyubashenko and the second named author. We consider separately the cases of $q$ generic and $q$ at roots of unity. Some of the representations have no classical analog even for generic $q$. Some of the representations have no analog to the finite-dimensional representations of the quantised enveloping algebra $U_q(sl(2))$, while in those that do there are different matrix elements., Comment: 14 pages, plain-TEX file using input files harvmac.tex, amssym.def

We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group $GL_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: $U'_{g,h}$ (with three generators) and $U(Z)$ (with one generator). The subalgebra $U(Z)$ is a central Hopf subalgebra of $U_{g,h}$. The subalgebra $U'_{g,h}$ is not a Hopf subalgebra and its coalgebra structure depends on both parameters. We discuss also two one-parameter special cases: $g =h$ and $g=-h$. The subalgebra $U'_{h,h}$ is a Hopf algebra and coincides with the algebra introduced by Ohn as the dual of $SL_h(2)$. The subalgebra $U'_{-h,h}$ is isomorphic to $U(sl(2))$ as an algebra but has a nontrivial coalgebra structure and again is not a Hopf subalgebra of $U_{-h,h}$., Comment: plain TeX with harvmac, 16 pages, added Appendix implementing the ACC nonlinear map

A new approach to the theory of polynomial solutions of q - difference equations is proposed. The approach is based on the representation theory of simple Lie algebras and their q - deformations and is presented here for U_q(sl(n)). First a q - difference realization of U_q(sl(n)) in terms of n(n-1)/2 commuting variables and depending on n-1 complex representation parameters r_i, is constructed. From this realization lowest weight modules (LWM) are obtained which are studied in detail for the case n=3 (the well known n=2 case is also recovered). All reducible LWM are found and the polynomial bases of their invariant irreducible subrepresentations are explicitly given. This also gives a classification of the quasi-exactly solvable operators in the present setting. The invariant subspaces are obtained as solutions of certain invariant q - difference equations, i.e., these are kernels of invariant q - difference operators, which are also explicitly given. Such operators were not used until now in the theory of polynomial solutions. Finally the states in all subrepresentations are depicted graphically via the so called Newton diagrams., Comment: uuencoded Z-compressed .tar file containing two ps files

We construct representations $\hat\pi_{\br}$ of the quantum algebra $U_q(sl(n))$ labelled by $n-1$ complex numbers $r_i$ and acting in the space of formal power series of $n(n-1)/2$ non-commuting variables. These variables generate a flag manifold of the matrix quantum group $SL_q(n)$ which is dual to $U_q(sl(n))$. The conditions for reducibility of $\hat\pi_{\br}$ and the procedure for the construction of the $q$ - difference intertwining operators are given. The representations and $q$ - difference intertwining operators are given in the most explicit form for $n=3$. In the Note Added some general results for arbitrary $n$ are given., Comment: 24 pages; v2 includes Note Added published in J.Phys. A27 (1994), references updated

The positive-energy unitary irreducible representations of the $q$-deformed conformal algebra ${\cal C}_q = {\cal U}_q(su(2,2))$ are obtained by appropriate deformation of the classical ones. When the deformation parameter $q$ is $N$-th root of unity, all these unitary representations become finite-dimensional. For this case we discuss in some detail the massless representations, which are also irreducible representations of the $q$-deformed Poincar\'e subalgebra of ${\cal C}_q$. Generically, their dimensions are smaller than the corresponding finite-dimensional non-unitary representation of $su(2,2)$, except when $N=2$, $h=0$ and $N = 2 \vert h\vert +1$, where $h$ is the helicity of the representations. The latter cases include the fundamental representations with $h = \pm 1/2$., Comment: 10 pages, plain-TEX, preprint IC/92/187, SISSA 137/92/FM, UTS-DFT-22-92, (July 1992), Phys. Lett. 302B (1993) 215

We consider BRST quantized 2D gravity coupled to conformal matter with arbitrary central charge $c^M = c(p,q) < 1$ in the conformal gauge. We apply a Lian-Zuckerman $SO(2,\bbc)$ ($(p,q)$ - dependent) rotation to Witten's $c^M = 1$ chiral ground ring. We show that the ring structure generated by the (relative BRST cohomology) discrete states in the (matter $\otimes$ Liouville $\otimes$ ghosts) Fock module may be obtained by this rotation. We give also explicit formulae for the discrete states. For some of them we use new formulae for $c <1$ Fock modules singular vectors which we present in terms of Schur polynomials generalizing the $c=1$ expressions of Goldstone, while the rest of the discrete states we obtain by finding the proper $SO(2,\bbc)$ rotation. Our formulae give the extra physical states (arising from the relative BRST cohomology) on the boundaries of the $p \times q$ rectangles of the conformal lattice and thus all such states in $(1,q)$ or $(p,1)$ models., Comment: 15 pages