Prime Ideals in Quantum Algebras

The central objects of study in this thesis are quantized coordinate algebras. These algebras originated in the 1980s in the work of Drinfeld and Jumbo and are noncommutative analogues of coordinate rings of algebraic varieties. The organic nature by which these algebras arose is of great interest to algebraists. In particular, investigating ring theoretic properties of these noncommutative algebras in comparison to the properties already known about their classical (commutative) counterparts proves to be a fruitful process. The prime spectrum of an algebra has always been seen as an important key to understanding its fundamental structure. The search for prime spectra is a central focus of this thesis. Our focus is mainly on Quantum Grassmannian subalgebras of quantized coordinate rings of Matrices of size m × n (denoted Oq(Mm,n)). Quantum Grassmannians of size m × n are denoted Gq(m, n) and are the subalgebras generated by the maximal quantum minors of Oq(Mm,n). In Chapter 2 we look at the simplest interesting case, namely the 2 × 4 Quantum Grassmannian (Gq(2, 4)), and we identify the H-primes and automorphism group of this algebra. Chapter 3 begins with a very important result concerning the dehomogenisation isomorphism linking Gq(m,n) and Oq(Mm,n−m). This result is applied to help to identify H-prime spectra of Quantum Grassmannians. Chapter 4 focuses on identifying the number of H-prime ideals in the 2×n Quantum Grassmannian. We show the link between Cauchon fillings of subpartitions and H-prime ideals. In Chapter 5, we look at methods of ordering the generating elements of Quantum Grassmannians and prove the result that Quantum Grassmannians are Quantum Graded Algebras with a Straightening Law is maintained on using one of these alternative orderings. Chapter 6 looks at the Poisson structure on the commutative coordinate ring, G(2, 4) encoded by the noncommutative quantized algebra Gq(2, 4). We describe the symplectic ideals of G(2, 4) based on this structure. Finally in Chapter 7, we present an analysis of the 2 × 2 Reflection Equation Algebra and its primes. This algebra is obtained from the quantized coordinate ring of 2 × 2 matrices, Oq(M2,2).