We present an uniform construction of the solution to the Yang-Baxter equation with the symmetry algebra sℓ(2) and its deformations: the q-deformation and the elliptic deformation or Sklyanin algebra. The R-operator acting in the tensor product of two representations of the symmetry algebra with arbitrary spins ℓ 1 and ℓ 2 is built in terms of products of three basic operators S 1 , S 2 , S 3 which are constructed explicitly. They have the simple meaning of representing elementary permutations of the symmetric group S 4 , the permutation group of the four parameters entering the RLL-relation.1

Abstract. We find solutions of the Yang-Baxter equation acting on tensor product of arbitrary representations of the superalgebra sℓ(2|1). Based on these solutions we construct the local Hamiltonians for integrable homogeneous periodic chains and open chains.

Abstract.We find a new 4 × 4 solution to the osp q (1|2)-invariant Yang-Baxter equation with simple dependence on the spectral parameter and propose 2 × 2 matrix expressions for the corresponding Lax operator. The general inhomogeneous universal spectral-parameter dependent R-matrix is derived. It is proven, that there are two independent solutions to the homogeneous osp q (1|2)-invariant YBE, defined on the fundamental three dimensional representations. One of them is the particular case of the universal matrix, while the second one does not admit generalization to the higher dimensional cases. Also the 3 × 3 matrix expression of the Lax operator is found, which have a well defined limit at q → 1.

The general structure of trace anomaly, suggested recently by Deser and Shwimmer, is argued to be the consequence of the Wess-Zumino consistency condition. The response of partition function on a finite Weyl transformation, which is connected with the cocycles of the Weyl group in d = 2k dimensions is considered, and explicit answers for d = 4, 6 are obtained. Particularly, it is shown, that addition of the special combination of the local counterterms leads to the simple form of that cocycle, quadratic over Weyl field σ, i.e. the form, similar to the two-dimensional Lioville action. This form also establishes the connection of the cocycles with conformal-invariant operators of order d and zero weight. Beside that, the general rule for transformation of that cocycles into the cocycles of diffeomorphisms group is presented. *

The spin1 Uimin-Lai-Sutherland (ULS) isotropic chain model is expressed in terms of fermions and the equivalence of the fermionic representation to the supersymmetric t − J model is established directly at the level of Hamiltonians.The spin-S ULS model is fermionized and the Hamiltonian of the corresponding generalisation of the t − J model is written down.