Yangians and its doubles of classical Lie superalgebras: representations theory and formula of universal R-matrix

Abstract:
The Yangian Y(g) of classical Lie Superalgebra g (or superYangian) is introduced as a specialization of quantization of the bisuperalgebra of currents g[u] with cosuperalgebra structure defined by rational r-matrix. The Yangians of simple Lie algebras play the role of the dynamical symmetries in quantum field theories and are used in investigation of integrable models of quantum mechanics and statistical physics. Tensor products of finite-dimensional representations of the Yangians produce rational solutions of the Yang-Baxter equations. The theory of Yangians for Lie superalgebras is less developed than the one for Lie algebras. In this report we investigate the superalgebra and cosuperalgebra structures of superYangians, describe finite-dimensional representations of superYangians of basic Lie superalgebras, introduce double of Yangian for basic Lie superalgebras and compute universal R-matrix for double of superYangian for superalgebras type A(m,n).

First, we introduce the analogue of the ``new system generators'' for superYangians and describe cosuperalgebra structure. We formulate PBW-theorem for basic Lie superalgebras and prove some results about cosuperalgebra structure. After them we investigate representations of superYangians. The finite-dimensional representations of Yangians of simple Lie algebras were described by V.Drinfel'd in [2]. The Yangians of Lie superalgebras of A(m,n) type were described in [3] and its representations were described in [4]. In this report we generalize this result from [4] on the case of basic Lie superalgebras. Namely, we introduce the Drinfel'd polynomials {P^d_i} for highest weight representations of superYangian Y(g) which are analogues of highest weights for Yangians and formulate theorem which describes irreducible finite-dimensional representations of superYangians in terms of Drinfel'd polynomials. This result is generalization of corresponding results from [2, 4, 5]. We define double structure for superYangian and compute pairing formulas in doubles, using comultiplication formula following [6]. We use action degenerate Hecke algebra for definition base in double and computing pairing. We derive multiplicative formula for universal R-matrix from this formulas.

References:

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