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 quant
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