On $q$-analogues of certain prehomogeneous vector spaces: comparison of several approaches

There exist several approaches to constructing $q$-analogues of prehomogeneous vector spaces of commutative parabolic type. In the present paper we compare three approaches developed by H. P. Jakobsen, T. Tanisaki et al., and L. Vaksman et al. Within framework of these three approaches the following problem is solved: a $q$-analogue of the algebra ${\mathbb C}[V]$ of holomorphic polynomials on an arbitrary irreducible prehomogeneous vector space $V$ (of commutative parabolic type) is constructed, and, moreover, the corresponding (non-commutative) algebra is endowed with a structure of $U$-module algebra with $U$ being certain quantum universal enveloping algebra. We prove that the three $q$-analogues of ${\mathbb C}[V]$ are isomorphic as $U$-module algebras. For the sake of simplicity we consider only the case when $V$ is the space of $2\times2$ complex matrices. But we present such proof which is transferable to the case of an arbitrary irreducible prehomogeneous vector space of commutative parabolic type.