An Introduction to Decomposition Classes

Decomposition Classes provide a natural way of partitioning a Lie algebra into finitely many pieces, collecting together adjoint orbits with similar Jordan decompositions. The current literature surrounding these tends to only cover certain cases -- such as in characteristic zero, or under the Standard Hypotheses. Building on the prior work of Borho-Kraft, Spaltenstein, Premet-Stewart and Ambrosio, we have managed to adapt many of the useful properties of decomposition classes to work in greater generality.

 

This talk will introduce the concept of Decomposition Classes, beginning with an illustrative example of 4-by-4 matrices over the complex numbers. We will then generalise this to the Lie algebras of connected reductive algebraic groups -- defined over arbitrary algebraically closed fields. After listing some general properties of Decomposition Classes and their closures, we will investigate structural differences across semisimple algebraic groups of type A_3, for different characteristics.