Bitableaux bases of the quantum coordinate algebra of a semisimple group

We extend the Standard Basis Theorem of Rota et al. to the setting of quantum symmetrizable Kac–Moody algebras. In particular, we obtain a procedure to give a presentation of the quantum coordinate algebra of any semisimple group, for generic q. More precisely, given any integrable module V of a quantum symmetrizable Kac–Moody algebra , we obtain a generating set of the ideal of relations among the matrix coefficients of V, and we give an upper bound for the degrees of these polynomials. Our approach is based on the theory of crystal bases and Littelmannʹs generalization of the plactic algebra.