Prime Ideals in Certain Quantum Determinantal Rings

The ideal I 1 generated by the 2 2 quantum minors in the coordinate algebra of quantum matrices, O q (M m;n (k)), is investigated. Analogues of the First and Second Fundamental Theorems of Invariant Theory are proved. In particular, it is shown that I 1 is a completely prime ideal, that is, O q (M m;n (k))=I 1 is an integral domain, and that O q (M m;n (k))=I 1 is the ring of coinvariants of a coaction of kx; x ?1 ] on O q (k m) O q (k n), a tensor product of two quantum aane spaces. There is a natural torus action on O q (M m;n (k))=I 1 induced by an (m + n)-torus action on O q (M m;n (k)). We identify the invariant prime ideals for this action and deduce consequences for the prime spectrum of O q (M m;n (k))=I 1 .