MULTIPLICATION FORMULAS AND SEMISIMPLICITY FOR -SCHUR SUPERALGEBRAS

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for the $q$-Schur superalgebras. This gives a combinatorialization of the relative norm approach developed in Du and Gu (A realization of the quantum supergroup$\mathbf{U}(\mathfrak{g}\mathfrak{l}_{m|n})$, J. Algebra 404 (2014), 60–99). We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for $q$-Schur superalgebras. We also construct infinitesimal and little $q$-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

Bicovariant differential calculus on the quantum superspace ℝq(1|2)

Super-Hopf algebra structure on the function algebra on the extended quantum superspace has been defined. It is given a bicovariant differential calculus on the superspace. The corresponding (quantum) Lie superalgebra of vector fields and its Hopf algebra structure are obtained. The dual Hopf algebra is explicitly constructed. A new quantum supergroup that is the symmetry group of the differential calculus is found.

The Hopf algebra structure of the h-deformed Z3-graded quantum supergroup GLh,j(1|1)

In this work, we define a new proper singular [Formula: see text] matrix to construct a Z3-graded calculus on the [Formula: see text]-deformed quantum superplane. Using the obtained calculus, we construct a new [Formula: see text]-deformed Z3-graded quantum supergroup and give some features of it. Finally, we build up the Hopf algebra structure of this supergroup.

HOPF ALGEBRA STRUCTURE OF (2+1)-DIMENSIONAL QUANTUM SUPERSPACE, ITS DUAL AND ITS DIFFERENTIAL CALCULUS

We consider a (2+1)-dimensional quantum superspace which has noncommuting coordinates in Manin sense and it was shown that this space has a Hopf algebra structure, i.e. the quantum supergroup, when it is extended by the inverse of the bosonic variable. Differential structures on this space were given by constructing the differential calculus in the sense of Woronowicz. Thus, we deduce that the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is non-cocommutative q-deformed. Finally, dual Hopf superalgebra was given.