Splinter Group:
Ring Theory

Organizer: Cora Stack (ITT Dublin)

All talks will be held in the McMunn Lecture Theatre.

Tuesday, April 7 2009

14:15 - 14:55

Cora Stack (ITT Dublin):
Structure of Nilpotent Algebras

This talk will consist of a survey of recent results from 1996 - 2007 on the structure theory of nilpotent algebras. It will survey recent work of Stack, Amberg, Kazarin and Soules among others.

14:55 - 15:35

Vladimir V. Kisil (University of Leeds):
Induced representations of SL(2,\Bbb R)

Starting from the Iwasawa decomposition of SL(2,\Bbb R) we review the construction of induced representations and Kirillov's orbit method. There are various links between discrete, principal, complementary series and hypercomplex extensions of real numbers. Moreover induced representations are closely connected to analytic function theories.

15:35 - 16:15

Miroslav Korbelář (Charles University in Prague):
2-generated algebras and Eggert's conjecture

Let A be a commutative nilpotent finitely-dimensional algebra over a field F of characteristic p > 0. A conjecture of Eggert(1971) says that p dim A^{(p)}\leq dim A, where A^{(p)} is the subalgebra of A generated by elements a^p, a \in A. We show that the conjecture holds if A^{(p)} is at most 2-generated. We present a type of canonical bases for 2-generated nilpotent algebras and give a complete characterization of their polynomial presentation with respect to these canonical bases.

16:35 - 17:15

Marcus Greferath (UCD):
Monomial Extension of Isometries between Ring-Linear Codes

This talk reports on ongoing work regarding the extension of isometries between ring-linear codes. It has been known for more than a decade that Hamming isometries (and also homogeneous isometries) between linear codes over finite Frobenius rings allow for a monomial extension to the ambient space. Not much is known however for the case where the weight function on the ring is more general. We will tackle this problem in our talk and present a solution for integer residue rings. A result for more general rings seems to be within reach, but has not been proved yet.

Wednesday, April 8 2009

14:15 - 14:55

Judith Millar (Queen's University Belfast):
K-Theory of Azumaya algebras

The concept of a central simple algebra over a field is generalised by an Azumaya algebra over a commutative ring.
  We will look at the relationship between the K-theory of an Azumaya algebra and the K-theory of its centre.

14:55 - 15:35

Eimear Byrne (UCD):
Measuring optimality of codes over finite rings

The important discovery in the 1990s that several families of good binary nonlinear codes have a \Bbb Z_4-linear representation reinvigorated the study of algebraic codes over finite rings. By now several papers have been written on the subject, on topics ranging from MacWilliams' extension and duality theorems, to code optimality to decoding. A linear [n, d] code C over a finite ring R is a submodule of \mbox{}_R R^n such that the minimum weight of any word of C for a given weight function is d. A fundamental question in coding theory is to determine the value of B_R (n, d), the maximal number of codewords a linear [n, d] code over R can have. For this reason upper bounds on B_R (n, d) are sought. A new weight function (which may be viewed as a natural generalisation of the Hamming weight for codes over finite rings), namely the homogeneous weight has emerged as useful in the context of finite rings. Examples of homogeneous weights include the Hamming weight on finite fields and the Lee weight on \Bbb Z_4. In this talk we discuss bounds on codes over finite rings for the homogeneous weight, including generalisations of the classical Plotkin, Elias, linear programming and Singleton bounds. We give some examples of codes meeting these bounds.

15:35 - 16:15

Leo Creedon (Sligo Institute of Technology):
Unitary Units of the Modular Quaternion Group Algebra

The structure of the unitary unit group of the modular quaternion group algebra \Bbb F_{2^k}Q_8 is described as a Hamiltonian group.

16:35 - 17:15

David Pauksztello (University of Leeds):
Co-t-structures and triangulated categories

In this talk we introduce the new notion of a co-t-structure on a triangulated category which is almost dual to a t-structure. Co-t-structures share many similarities with t-structures but in some ways they behave rather differently. We illustrate the notion with an example of a cochain differential graded algebra.