DUALITY FOR MULTIPARAMETER QUANTUM DEFORMATION OF THE SUPERGROUP GL(m/n)

Abstract

We find the Hopf superalgebra [Formula: see text], which is in duality with the multiparameter quantum deformation GL u q (m/n) of the supergroup GL (m/n). Naturally [Formula: see text] is a multiparameter deformation of the superalgebra U ( gl (m/n)). We show that as a commutation algebra we have the classical structure, namely a split into two subalgebras: [Formula: see text], where [Formula: see text] is isomorphic to the standard one-parameter deformation U u( sl (m/n)), and [Formula: see text] is central in [Formula: see text] for m ≠ n. However, as a coalgebra [Formula: see text] cannot be split in this way, as only [Formula: see text] is a Hopf subalgebra, while [Formula: see text] is not a Hopf subalgebra unless m = n = 1 or some special relations between the parameters exist. These special relations are established and used to obtain explicit multiparameter Hopf superalgebra deformations of U ( sl (m/n)).