We address the problem of classifying of irreducible Gelfand-Tsetlin modules for gl(m|n) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. We also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand-Tsetlin modules which includes all essentially typical and covariant tensor finite dimensional modules. In the quasi typical case new irreducible representations are infinite dimensional gl(m|n)-modules which are isomorphic to the parabolically induced (Kac) modules. Preliminaries2.1. Weight modules. A Z 2 -graded vector space g = g0 ⊕ g1 with even bracket [•, •] : g ⊗ g → g is a Lie superalgebra iff the following conditions hold [a, b] = −(−1) p(a)p(b) [b, a]; Berkeley

For an admissible affine vertex algebra V k (g) of type A, we describe a new family of relaxed highest weight representations of V k (g). They are simple quotients of representations of the affine Kac-Moody algebra g induced from the following g-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from sl 2 ; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from sl 3 ; 3) all simple Gelfand-Tsetlin modules over sl 3 . This in particular gives the classification of all simple positive energy weight representations of V k (g) with finite dimensional weight spaces for g = sl 3 .

We provide a classification and an explicit realization of all irreducible Gelfand-Tsetlin modules of the complex Lie algebra sl(3). The realization of these modules uses regular and derivative Gelfand-Tsetlin tableaux. In particular, we list all simple Gelfand-Tsetlin sl(3)-modules with infinitedimensional weight spaces. Also, we express all simple Gelfand-Tsetlin sl(3)modules as subquotionets of localized Gelfand-Tsetlin E 21 -injective modules.MSC 2010 Classification: 16G99, 17B10.

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Abstract. Consider a Hölder continuous potential φ defined on the full shift A N , where A is a finite alphabet. Let X ⊂ A N be a specified sofic subshift. It is well-known that there is a unique Gibbs measure µ φ on X associated to φ. Besides, there is a natural nested sequence of subshifts of finite type (Xm) converging to the sofic subshift X. To this sequence we can associate a sequence of Gibbs measures (µ m φ ). In this paper, we prove that these measures weakly converge at exponential speed to µ φ ( in the classical distance metrizing weak topology). We also establish a strong mixing property (ensuring weak Bernoullicity) of µ φ . Finally, we prove that the measure-theoretic entropy of µ m φ converges to the one of µ φ exponentially fast. We indicate how to extend our results to more general subshifts and potentials. We stress that we use basic algebraic tools (contractive properties of iterated matrices) and symbolic dynamics.