On the 2D zero modes' algebra of the SU(n) WZNW model

A quantum group covariant extension of the chiral parts of the Wess-Zumino-Novikov-Witten model on a compact Lie group G gives rise to two matrix algebras with non-commutative entries. These are generated by "chiral zero modes" which combine in the 2D model into "Q-operators" which encode information about the internal symmetry and the fusion ring. We review earlier results about the SU(n) WZNW Q-algebra and its Fock representation for n=2 and display the first steps towards their generalization to higher n.Comment: 10 pages, Talk presented by L.H. at the International Workshop LT10 (17-23 June 2013, Varna, Bulgaria

Fusion Rings Related to Affine Weyl Groups

The construction of the fusion ring of a quasi-rational CFT based on $\hat{sl}(3)_k$ at generic level $k\not \in {\Bbb Q}$ is reviewed. It is a commutative ring generated by formal characters, elements in the group ring ${\Bbb Z}[\tilde{W}]$ of the extended affine Weyl group $\tilde{W}$ of $\hat{sl}(3)_k$. Some partial results towards the $\hat{sl}(4)_k$ generalisation of this character ring are presented.Comment: 13 pages; two figures. Talk at ``Lie Theory and Its Applications in Physics III'', Clausthal, 11-14 July, 1999, to appear in the Proceedings, eds. H.-D. Doebner et a

Chiral zero modes of the SU(n) Wess-Zumino-Novikov-Witten model

We define the chiral zero modes' phase space of the G=SU(n) Wess-Zumino-Novikov-Witten model as an (n-1)(n+2)-dimensional manifold M_q equipped with a symplectic form involving a special 2-form - the Wess-Zumino (WZ) term - which depends on the monodromy M. This classical system exhibits a Poisson-Lie symmetry that evolves upon quantization into an U_q(sl_n) symmetry for q a primitive even root of 1. For each constant solution of the classical Yang-Baxter equation we write down explicitly a corresponding WZ term and invert the symplectic form thus computing the Poisson bivector of the system. The resulting Poisson brackets appear as the classical counterpart of the exchange relations of the quantum matrix algebra studied previously. We argue that it is advantageous to equate the determinant D of the zero modes' matrix to a pseudoinvariant under permutations q-polynomial in the SU(n) weights, rather than to adopt the familiar convention D=1.Comment: 30 pages, LaTeX, uses amsfonts; v.2 - small corrections, Appendix and a reference added; v.3 - amended version for J. Phys.

An Extension of the Character Ring of sl(3) and Its Quantisation

We construct a commutative ring with identity which extends the ring of characters of finite dimensional representations of sl(3). It is generated by characters with values in the group ring $Z[\tilde{W}]$ of the extended affine Weyl group of $\hat{sl}(3)_k$ at $k\not \in Q$. The `quantised' version at rational level $k+3=3/p$ realises the fusion rules of a WZW conformal field theory based on admissible representations of $\hat{sl}(3)_k$.Comment: contains two TeX files: main file using harvmac.tex, amssym.def, amssym.tex, 35p.; file with figures using XY-pic package, 4p; v2: minor corrections, Note adde

Fusion rules for admissible representations of affine algebras: the case of A2(1)A_2^{(1)}

We derive the fusion rules for a basic series of admissible representations of $\hat{sl}(3)$ at fractional level $3/p-3$. The formulae admit an interpretation in terms of the affine Weyl group introduced by Kac and Wakimoto. It replaces the ordinary affine Weyl group in the analogous formula for the fusion rules multiplicities of integrable representations. Elements of the representation theory of a hidden finite dimensional graded algebra behind the admissible representations are briefly discussed.Comment: containing two TEX files: main file using input files harvmac.tex, amssym.def, amssym.tex, 19p.; file with figures using XY-pic package, 6p. Correction in the definition of general shifted weight diagra

Influence of disordered porous media in the anomalous properties of a simple water model

The thermodynamic, dynamic and structural behavior of a water-like system confined in a matrix is analyzed for increasing confining geometries. The liquid is modeled by a two dimensional associating lattice gas model that exhibits density and diffusion anomalies, in similarity to the anomalies present in liquid water. The matrix is a triangular lattice in which fixed obstacles impose restrictions to the occupation of the particles. We show that obstacules shortens all lines, including the phase coexistence, the critical and the anomalous lines. The inclusion of a very dense matrix not only suppress the anomalies but also the liquid-liquid critical point