Department of Mathematical Sciences
Florida Atlantic University
Boca Raton, Florida 33431-0991

# Boca Raton Algebra Seminar

Spring 2001

All lectures take place in Science and Engineering, room 215.  Everybody is invited to participate.

Monday, May 14, 10:30 a.m. S&E 215

### Clans IX:  A Review of Ballantines Classification

Lee Klingler

Abstract.  The indecomposable representations corresponding to a symmetric string with associated quadratic polynomials  q_b(T)  and  q_c(T)  are in one-one correspondence to the indecomposable modules over the  algebra

A=k < S , T >  /  ( q_b(S) , q_c(T) )

In this talk we will review Ballantines results about  A -modules.

Wednesday, May 8, 2 p.m. S&E 215 --- POSTPONED ---

### Isomorphism Invariants for 2-Dimensional Algebras

Jack Freeman

Abstract.   Corresponding to a (commutative and associative) 2-dimensional algebra  A  over a field  k , there is an invariant way of defining harmonic functions  u ( x_1 , x_2 )  and their  conjugates  v ( x_1 , x_2 ) .   A vector subspace  U_2 ( x_1 , x_2 )  of the homogeneous symmetric functions of degree 2 characterizes  A  to within isomorphism.

Tuesday, May 1, 1 p.m., S&E 215

### Why We Multiply As We Do, Part II: Applications

Jack Freeman

See the abstract of the previous talk.

Tuesday, April 24, 1 p.m., S&E 215

### Why We Multiply As We Do

Jack Freeman

Abstract.  Starting with an algebra  over  a field  k , there are lots of ways of defining a multiplication on  A x A .  Among the most familiar are the direct product, the defninition of which yield the complex numbers or the dual numbers from the reals, or quaternions from complex numbers.  Thinking of these definitions as procedures with input  A , they are, save for the last example, generic in that the only data used are from the field  k . (In the case of the quaternions  C x from  over  one needs the involution  \lambda(z) = \bar on too.  That is, beyond  R , one needs something from  Aut(C)).

In the generic cases, the multiplication is determined by a pair of  2 x 2 -matrices over  k .  Using this we completety catalog the generic multiplications on  A x A  which inherit  (seperately) either commutativity or associativity from  A .

There is a surprising profusion of these - some, I believe, both interesting and new.  An example being

( x_1 , x_2 ) . ( y_1 , y_2 )  =  ( x_1 y_1 + \alpha( x_1 y_2 + x_2 y_1) + x_2 y_2 , x_2 y_2 )

where the golden ratio  (\sqrt(5)+1)/2  and its conjugate  (\sqrt(5)-1)/2   are the only possible values for  \alpha .

We have also established some partial fuzzy boundaries among geometric isomorphism classes of these algebras.  It would be nice to find some numerical invariants, based directly on the pair of matrices, to sharpen up these boundaries.  No success yet.

Tuesday, April 17, 1 p.m., S&E 215

### Clans - Part IV

Markus Schmidemeier

In this talk we study elementary properties of the category of representations of a clan and introduce constructions for indecomposable representations.

Tuesday, April 10, 1 p.m., S&E 215

### Clans  - Part III

Lee Klingler

In this third part of our series of lectures on clans we introduce self-reproducing matrix problems and show that they form a clan. By doing so, we hope to translate classification results for matrix problems into corresponding results for clans, and conversely.

Tuesday, April 3, 1 p.m., S&E 215

### Clans  -  Part II

Lee Klingler

Many classification problems --- for example: representations of tame posets, self-reproducing matrix problems, representations of tame quivers --- can be expressed in the language of clans.  And indeed, by writing a classification problem as a clan, we gain insight in the structure of the category under investigation.

Tuesday, March 27, 1 p.m., S&E 215

### Clans

Lee Klingler

Abstract.  In his manuscript, Functorial Filtations II: Clans and the Gelfand Problem, W. Crawley Boevey introduced a class of matrix problems which he called "clans" in order to solve certain problems of tame representation type.  Our interest in clans stems from the fact that the indecomposable representations of a clan can be described in terms of symmetric and asymmetric strings and bands, of exactly the type that describe indecomposable finitely generated modules over unsplit Dedekind-like rings. Surely there is a connection.

In this (first) talk I shall define clans and their representations and look at a number of "natural" examples of clans.

Tuesday, March 20, 1 p.m., S&E 215

### What is a PID ?  -  Part III

Fred Richman

This is definitely the final go at this topic. Please see the previous abstract which I really didn't get to last time. The content is mostly definitions and examples.

Tuesday, March 6, 1 p.m., S&E 215

### What is a PID ?  -  Part II

Fred Richman

Various candidates for the title role are, in addition to Noetherian Bezout domains, are:

• Euclidean domains
• Normed Bezout domains
• Bounded Bezout domains
• Well-founded Bezout domains

• These are in decreasing order of strength. I might also get to the even weaker notions of adequate Bezout domains and subadequate Bezout domains. These are all elementary divisor rings, that is, you can diagonalize matrices over them.

Tuesday, February 27, 1 p.m., S&E 215

### What is a PID ?

Fred Richman

I have a file, dated 16 March 1999, entitled "Dedekind domains: a constructive development without valuations or choice" which I would like figure out. I've decided to take advantage of the lull in the algebra seminar to dust it off. I'll start by talking about principal ideal domains because I understand those a lot better and because  they provide a less complicated illustration of some of the  constructive problems involved.

Tuesday, January 30, 1 p.m., S&E 215

### Submodules of modules over the polynomial ring  k[T]

Markus Schmidmeier

Abstract.  Denote by  S  the category of pairs  (U, V)  consisting of a finitely generated k[T]-module V  and a submodule  U  of  V . There is no hope to get a full classification of all indecomposable pairs in  S, up to isomorphism,  so let us consider the full subcategories  S(m,n)  of  S, for m < n,  consisting of the pairs (U, V)  such that

U  is annihilated by  T^m  and
V  is annihilated by  T^n.

The following theorem, which is part of joint work with Claus M. Ringel, generalizes related results in the recent book by David Arnold.

Theorem.  Let  m < be positive integers.

• The category  S(m,n)  is representation finite if  is at most  5, if m  is at most  2, or if  (m,n) = (3,6).
• The category  S(m,n)  has tame infinite representation type in case  (m,n)  is one of the pairs  (4,6), (5,6), or (3,7).
• Otherwise,  S(m,n)  is of wild representation type in the sense that there are two parameter families of indecomposable objects of a  certain dimension type.
In the case that  S(m,n)  is of finite or tame representation type, a full classification of the indecomposable pairs  (U, V)  can be given. Surprisingly, if  m = 2, the shape of the Auslander-Reiten quiver depends on whether  n  is even or odd...

Tuesday, 23 January, 1 p.m., S&E 215

### Groups, t-Designs and Large Sets

Spyros Magliveras

Algebra Meetings:
Weekend Algebra Conference at the University of Southern Mississippi (April 20-22, Hattisburg, Mississippi)
Joint Meeting of the Belgian and German Mathematical Societies (June 6-10, 2001, Liege Belgium, with a representation theory section)
5th Budapest-Chemnitz-Prague-Torun Algebra Symposion (June 14-16, 2001, Budapest)
Honolulu Conference on Abelian Groups and Modules (July 25 - August 1, 2001)
Homological Conjectures for Finite Dimensional Algebras (Summerschool, August 12-19, Nordfjordeid, Norway)
For further meetings on the Representation Theory of Finite Dimensional Algebras see FDLIST.

Current Program: Algebra-Cryptology Seminar