Finite dimensional simple modules of (q,Q)-current algebras

The ( q , Q ) -current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic q-Schur algebras. In this paper, we study the ( q , Q ) -current algebra U q ( sl n 〈 Q 〉 [ x ] ) associated with the special linear Lie algebra sl n . In particular, we classify finite dimensional simple U q ( sl n 〈 Q 〉 [ x ] ) -modules.