Realizations of Quantum Hom-Spaces, Invariant Theory, and Quantum Determinantal Ideals

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