Realizations of quantum hom-spaces, invariant theory and quantum determinantal ideals

Abstract

For a Hecke operator $R$, one defines the matrix bialgebra $\E_R$, which is considered as the function algebra on the quantum space of endomorphisms of the quantum space associated to $R$. One generalizes this notion, defining the function algebra $\M_{RS}$ on the quantum space of homomorphisms of two quantum spaces associated to two Hecke operators $R$ and $S$ respectively. $\M_{RS}$ can be considered as a quantum analogue (or a deformation) of the function algebra on the variety of matrices of a certain degree. We provide two realiztions of $\M_{RS}$ as a quotient algebra and as a subalgebra of a tensor algebra, whence derive interesting informations about $\M_{RS}$, for instance the Koszul property, a formula for computing the Poincar\'e series. On $\M_{RS}$ coact the bialgebras $\E_R$ and $\E_S$. We study the two-sided ideals in $\M_{RS}$, invariant with respect to these actions, in particular, the determinantal ideals. We prove analogies of the fundamental theorems on invariant theory for these quantum groups and quantum hom-spaces.