Quantum Symmetric Algebras

The plus part U⁺ of a quantum group U_q(g) has been identified by M. Rosso with a subalgebra G_sym of an algebra G which is a quantized version of R. Ree's shuffle algebra. Rosso has shown that G_sym and G – and hence also Hopf algebras which are analogues of quantum groups – can be defined in a much wider context. In this paper we study one of Rosso's quantizations, which depends on a family of parameters t_ij. G_sym is determined by a family of matrices whose coefficients are polynomials in the t_ij. The determinants of the factorize into a number of irreducible polynomials, and our main Theorem 5.2a gives strong information on these factors. This can be regarded as a first step towards the (still very distant!) goal, the classification of the symmetric algebras G_sym which can be obtained by giving special values to the parameters t_ij.