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

For a Hecke operator R, one defines the matrix bialgebra E-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 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 analog (or a deformation) of the function algebra on the variety of matrices of a certain degree. We provide two realizations of M-RS as a quotient algebra and as a subalgebra of a tensor algebra, whence we derive interesting information about M-RS, for instance the Koszul property, a formula for computing the Poincare 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 of invariant theory for these quantum groups and quantum hom-spaccs. (C) 2002 Elsevier Science (USA).