Research of Edward S. Letzter

1. Summary. My prior and current research is primarily concerned with the internal structure, representation theory, and \noncommutative aane geometry" of certain classes of associative algebras. The examples directly involved arise from algebraic quantum groups, nite dimensional Lie superalgebras, and polycyclic-by-nite-groups; in particular, Hopf (super)algebras play an essential part. My focus has mainly been on prime and primitive ideal theory (see below), and the underlying goal is to understand how the primitive spectra of nitely generated noncommu-tative algebras compare with the maximal spectra of aane commutative algebras (i.e., with aane algebraic varieties). A similar aim is to understand the analogies between the primitive spectra of noncommutative Hopf algebras and aane algebraic groups. This latter objective is largely motivated by the role noncommutative Hopf algebras play in the theory of quantum groups. 2. Commutative Background. Algebraic geometry begins with a system f 1 = set forms an aane variety V in C n. The coordinate algebra O(V) is constructed as the quotient of the commutative polynomial ring C x 1 ; : : : ; x n ] modulo the polyno-mials vanishing on V, and Hilbert's Nullstellensatz states that V can be identiied with max O(V) (i.e., the set of maximal ideals, or maximal spectrum, of O(V)). Conversely, the assigment A 7 ! max(A) deenes an arrow reversing functor from the category of aane (i.e., nitely generated) commutative C-algebras A onto the category of C-aane varieties. In particular, morphisms of varieties correspond to algebra homomorphisms. Restricted to reduced algebras, max() deenes a category equivalence, with inverse O(). Under the Zariski topology, the closed subsets of V are the solutions to polynomial systems containing f 1 ; : : : ; f m. Equivalently, letting A be a commuta-tive aane C-algebra, the Zariski closed subsets of max A are those of the form V (J) = fm 2 max A j m Jg, for ideals J of A. From several points of view it is best to study max A within the (usually) larger set spec A of prime ideals, to which the Zariski topology naturally extends: The closed subsets are those of the form V (J) = fP 2 spec A j P Jg. A basic property of spec A is its catenarity: 0 are chains of prime ideals between P and P 0 that cannot be lengthened by the insertion of additional prime ideals, then m = n. A complex …