No abstract

A quantum symmetric pair is a quantization of the symmetric pair of universal enveloping algebras. Recent development suggests that most of the theory for quantum groups can be generalised to the setting of quantum symmetric pairs. In this paper, we study the ıquantum group at roots of 1. We generalize Lusztig's quantum Frobenius morphism in this new setting. We define the small ıquantum group and compute its dimension.

For a simple Lie superalgebra of type BDFG, we give explicit formulas for singular vectors in a Verma module of highest weight λ − ρ, which have weight s γ λ − ρ for certain positive non-isotropic roots γ. This implies the existence of a nonzero homomorphism between the corresponding Verma modules.2010 Mathematics Subject Classification. Primary 17B10.

A quantum covering group is an algebra with parameters q and π subject to π 2 = 1 and it admits an integral form; it specializes to the usual quantum group at π = 1 and to a quantum supergroup of anisotropic type at π = −1. In this paper we establish the Frobenius-Lusztig homomorphism and Lusztig-Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1. The specialization of these constructions at π = 1 recovers Lusztig's constructions for quantum groups at roots of 1.2010 Mathematics Subject Classification. Primary 17B37.