Department of Mathematical Sciences
Florida Atlantic University

Boca Raton Algebra Seminar

Fall 1999

Tuesday, 30 November, 3:00, S&E 215

Associated and weakly associated primes in polynomial rings

Jim Brewer

We will discuss these two notions of associated primes in the special case of polynomial rings. The ideas go back to Bourbaki in the late sixties. We will focus on a new proof by Carl Faith of the following old result: The restriction map from Ass(R[X])--->Ass(R) is a bijection. Thus, every associated prime of R[X] is extended from an associated prime of R.

Tuesday, 16 November, 3:00, S&E 215

A tale of four subspaces

Markus Schmidmeier and Fred Richman

Markus will talk about the four-subspace problem: classifying quadruples of subspaces V{1},V{2},V{3},V{4} of a vector space V. Fred will give an example of a quadruple that purports to show how difficult it is to classify subgroups of (infinite) p^6-bounded abelian groups.

Tuesday, 9 November, 3:00, S&E 215

Value-bounded valuated abelian p-groups

Fred Richman

These guys are abelian p-groups A with a descending chain of subgroups A = A(0), A(1), A(2), ... such that p(A(n)) is contained in A(n+1) for each n, and A(n) = 0 for some n. Classifying those A such that A(n) =
0 amounts to classifying subgroups of (p^n)-bounded abelian groups.

We can classify those A such that A(5) = 0. Each is a (possibly infinite) direct sum of indecomposables of which there are a finite number that can be described by small graphs. A satisfactory general description for A(6) = 0 would, among other things, allow us to classify (infinite dimensional) k[X]-modules where k is the p-element
field (so it can't be done).

Tuesday, 2 November, 3:00, S&E 215

Subgroups of finite abelian groups

Fred Richman

Finite abelian groups are completely understood in the sense that there is a simple structure theorem that describes all of them. To add some weight to the subject, I will describe why a square matrix with entries in a field the same as a finite abelian group. Thus finite abelian group theory encompasses linear algebra.

Subgroups of finite abelian groups, on the other hand, can be very complicated. Not as abelian groups in their own right, but as subgroups---that is, in how they relate to their containing group. The linear algebra analogues of these subgroups are invariant subspaces.

In this first talk, I'll give examples of how isomorphic subgroups of a group may be embedded in different ways, and introduce the main tool for their classification: valuated groups. We can't classify all of them, but there is a satisfactory classification for those contained in a group that is bounded by the fifth power of some prime.

Tuesday, 26 October, 3:00, S&E 215

Part III

Jim Brewer

Must the coefficients of the minimum polynomial of a matrix over a domain lie in that domain? This question leads to the notion of a weakly integrally closed domain, over which the answer is "yes" for 3-by-3 matrices. Certain subalgebras of k[t] are weakly integrally closed, as are rings consisting of quadratic algebraic numbers.

Tuesday, 12 October, 3:00 S&E 215

Part II

Jim Brewer

Must the coefficients of the minimum polynomial of a matrix over a domain lie in that domain? This question leads to the notion of a weakly integrally closed domain, over which the answer is "yes" for 3-by-3 matrices. Certain subalgebras of k[t] are weakly integrally closed, as are rings consisting of quadratic algebraic numbers.

Tuesday, 5 October, 3:00, S&E 215

Weakly integrally closed domains:  minimum polynomials of matrices

Jim Brewer

Abstract.

Must the coefficients of the minimum polynomial of a matrix over a domain lie in that domain? This question leads to the notion of a weakly integrally closed domain, over which the answer is "yes" for 3-by-3 matrices. Certain subalgebras of k[t] are weakly integrally closed, as are rings consisting of quadratic algebraic numbers.

3:00 Tuesday, 28 September, SE 215

Using Quantum Groups for the Construction of Knot Invariants II

Markus Schmidmeier

Abstract.   We describe the structure of the finite-dimensional simple modules over the quantum group U_q(sl(2)), and their R-matrices. From this data we obtain an explicit construction of the Jones-Conway polynomial for links.

Tuesday, 14 September 1999, 3:00, S&E 21

Using Quantum Groups for the Construction of Knot Invariants

Markus Schmidmeier

Abstract.  It is the aim of this very general talk to motivate the study of quantum groups and their modules by presenting the following famous application.

The data given by a quantum group, in particular the comultiplication, the counit, the antipode and the universal R-matrix, can be used to equip the category  M  of its modules with additional structure. This enables us to construct a family of nontrivial functors from the category of tangles into  M .  In case of the quantum group U_q(sl(2)) the values of these functors at knots or links can be interpreted as the coefficients of the Jones-Conway polynomial for links.

Tuesday, September 7,  at 3 p.m. in SE 215

Finite-length modules over thin Z-graded rings.

Markus Schmidmeier

Abstract. Quantized versions of classical noetherian rings, e. g. the quantized Weyl algebra, U_q(sl(2)), the quantum plane and the quantum 2-sphere, have enjoyed recent interest in ring theory, knot theory and theoretical physics. In the classification of their simple modules (Block, Rosenberg, Bavula), one reduces via factorization and localization to thin Z-graded rings. In particular, skew polynomial rings and Dedekind-like rings, as investigated by Arnold, Klingler, Laubenbacher and Levy, occur. It is the aim of this talk to introduce twisted versions of the notions of tame and wild that are adopted to the study of the category of finite-length modules over a large class of thin Z-graded rings.

Program of the Boca Raton Algebra Seminar:  Spring 2000, Current Program