Arithmetic lattices in unipotent algebraic groups

Computing in Unipotent and Reductive Algebraic Groups

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup of a split reductive group and show how this improves computation in the reductive group itself.

Constructing Arithmetic Subgroups of Unipotent Groups

Let G be a unipotent algebraic subgroup of some GLm(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G ∩ GLm(Z). This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.

Algebraic Groups and Arithmetic Groups

Algebraic Groups I. Unipotent radicals and reductivity

In class, we have proved the important fact that over any field k, a non-solvable connected reductive group containing a 1-dimensional split maximal k-torus is k-isomorphic to SL2 or PGL2. That proof relied on knowing that maximal tori remain maximal after a ground field extension to k, and so relies on Grothendieck’s theorem. But for algebraically closed fields there is no content to Grothendi...

Algebraic and Arithmetic Lattices . Part I 1

(1) Let L be a non empty reflexive transitive relational structure and x, y be elements of L. If x ≤ y, then compactbelow(x)⊆ compactbelow(y). (2) For every non empty reflexive relational structure L and for every element x of L holds compactbelow(x) is a subset of CompactSublatt(L). (3) For every relational structure L and for every relational substructure S of L holds every subset of S is a s...