  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 $\mathfrak{g}$.
