Typical irreducible characters of generalized quantum groups

We introduce the definition of the typical irreducible modules of the generalized quantum groups, and prove the Weyl–Kac-type formulas of their characters. As a by-product, we obtain the Weyl–Kac-type character formulas of the typical irreducible modules of the quantum superalgebras associated with the basic classical Lie superalgebras, which is explained in Introduction.

Two-parameter Quantum Superalgebras and PBW Theorem

A class of two-parameter quantum algebras [Formula: see text] is constructed. It is shown that [Formula: see text] is a Hopf superalgebra. Then the PBW basis of [Formula: see text] is described. For this purpose, some commutative relations of root vectors of [Formula: see text] are given.

LUSZTIG SYMMETRIES AND AUTOMORPHISMS OF QUANTUM SUPERALGEBRAS Uq(osp(1|2r))

We first construct algebra automorphisms Ti (1 ≤ i ≤ r) of ℤ2-graded Hopf algebra Uq( osp (1|2r)), which are called Lusztig's symmetries. It is shown that they satisfy braid group relations. The algebraic automorphisms of Uq( osp (1|2r)) are also described.