Factorizable Module Algebras

The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras which we call factorizable by generalizing the Gauss factorization of (square or rectangular) matrices. This class includes coordinate algebras of corresponding reductive groups $G$, their parabolic subgroups, basic affine spaces and many others. It turns out that tensor products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any $\mathfrak{g}$-module algebra. We also have quantum versions of all these constructions in the category of $U_q(\mathfrak{g})$-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra $U_q(\mathfrak{g}^*)$ of the dual Lie bialgebra $\mathfrak{g}^*$ of $\mathfrak{g}$.