Algebraic groups

We give an algorithm for working out the indecomposable direct summands in a Krull–Schmidt decomposition of a tensor product of two simple modules for SL3 in characteristics 2 and 3. It is shown that there is a finite family of modules such that every such indecomposable summand is expressible as a twisted tensor product of members of that family.

We survey some results on the structure of the groups which are definable in theories of fields involved in the applications of model theory to Diophantine geometry. We focus more particularly on separably closed fields of finite degree of imperfection.