Structure and Representations of the Quantum General Linear Supergroup

Abstract:

The structure and representations of the quantum general linear supergroup GL _q (m|n) are studied systematically by investigating the Hopf superalgebra G _q of its representative functions. G _q is factorized into $G_q^\pi G_q^{\bar \pi }$, and a Peter–Weyl basis is constructed for each factor. Parabolic induction for the quantum supergroup is developed. The underlying geometry of induced representations is discussed, and an analog of Frobenius reciprocity is obtained. A quantum Borel–Weil theorem is proven for the irreducible covariant and contravariant tensorial representations, and explicit realizations are given for classes of irredducible tensorial representatins in terms of sections of quantum super vector bundles over quantum projective superspaces.