Title:

Quantum Grassmannians and normal elements in Noetherian rings

The main idea running throughout the thesis is that of a normal element. An element in a ring is normal if the one sided ideal it generates is actually two sided. In the first half of the thesis (Chapters 2, 3 and 4), our aim is to study certain quantum coordinate algebras. We are particularly interested in two subalgebras of the coordinate ring of quantum matrices, namely quantum Grassmannians and quantum Flag varieties. A basis is constructed for each of these algebras, from which we calculate their GelfandKirillov dimensions. A well known result in the classical theory is that the dehomogenisation of the coordinate ring of the m x n Grassmannian at the 'rightmost' minor is isomorphic to the coordinate ring of the m x (n  m) matrices. The commutative notion of dehomogenisation does not immediately pass over to noncommutative theory. However, in the graded case, we show that it is possible to define noncommutative dehomogenisation at a regular normal homogeneous element, x, of degree 1 by considering a subring of the localisation of the ring at that element. The relationship between the prime spectrum of the original ring and that of the dehomogenisation is considered and a homeomorphism between the graded primes in the ring (not containing x) and a certain subset of primes in the dehomogenisation is constructed. Turning our attention back to quantum Grassmannians, we obtain the desired result, that the dehomogenisation of the quantum Grassmannians at the 'rightmost' minor is isomorphic to the quantum matrices. The smallest instructive example of a quantum Grassmannian is the 2 x 4 quantum Grassmannian, G_{q}(2, 4) and we restrict our attention to this case. By presenting the algebra as a factor ring of an iterated Ore extension, we see that G_{q}(2,4) is Auslander Gorenstein and Cohen Macaulay. We conjecture that this is also true in the general case. Finally, we consider the graded prime spectrum of G_{q}(2,4).
