Induced representations of the two parametric quantum deformation \(U_{pq}[\text{gl}(2/2)]\) (Q2738200)

A two-parameter quantum deformation \(U_{p,q}[gl(2|2)]\) of the enveloping algebra of the Lie superalgebra \(gl(2|2)\) is considered. A class of finite-dimensional representations is considered, induced from finite-dimensional representations of the even subalgebra. These representations are the deformed analogue of the induced modules constructed by \textit{V. G. Kac} [Lect. Notes Math. 676, 597-626 (1979; Zbl 0388.17002)], also known as the Kac-modules. NEWLINENEWLINENEWLINETwo different bases for these modules are constructed~: the first one is simply the induced basis; the second one is called the ``reduced basis'' and gives the direct decomposition with respect to irreducible representations of the even subalgebra. NEWLINENEWLINENEWLINEThen, transformation coefficients relating these two bases are given explicitly. It is pointed out that some of these coefficients are not simply \(pq\)-deformations of ordinary numbers, but seem to have a more complicated structure.