Classification of amalgams for non-spherical Kac-Moody groups (08rit130)


(Department of Mathematics and Statistics, Bowling Green State Univeristy)

Corneliu Hoffman (University of Birmingham)


Groups appear in various places in the world. They appear in such varied contexts as solutions to polynomial equations, solutions to differential equations (describing for instance the flow of oil through a pipeline), symmetries of time-space as in relativity theory, symmetries of crystals (useful in chemistry), and art, think for instance of Escher's regular tilings of the plane or the circle.

Groups can be understood in several ways. One way is to list all its elements and to write down a large multiplication table describing the product of any two of these elements. Another way is to analyze the space whose symmetries form that group. Unfortunately, both ways are not so effective in our study since we study groups that have infinitely many elements and are symmetries of infinite-dimensional spaces.
Instead we show that our groups can be understood through an amalgam. An amalgam for the group is just a small (in our case finite) set of elements of which we only know some products; it is chosen in such a way that we study the entire group just from this small piece. This is similar to a sudoku puzzle: usually the puzzle can be solved from the few numbers that are given. We study new unknown groups, find amalgams for them, and try to capture the essence of these amalgams using easily understood diagrams.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologí­a (CONACYT).