Levi Decomposition for Carpet Subgroups of Chevalley Groups Over a Field

On Levi subgroups and the Levi decomposition for groups definable in o-minimal structures

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an ominimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R ·S where R is the solvable radical of G. We also prove that any semisimple subalgebra of t...

Intermediate Subgroups in Chevalley Groups

Let G be a group and D be a subgroup of G. In this paper we are interested in the description of the lattice L(D; G) = fH j D H Gg of subgroups of G which contain D. We call subgroups H 2 L(D; G) subgroups intermediate between D and G. We are primarily interested in the parametrization of intermediate subgroups and their behaviour with respect to set-theoretic and group-theoretic operations lik...

gauss decomposition for chevalley groups, revisited

in the 1960's noboru iwahori and hideya matsumoto, eiichi abe and kazuo suzuki, and michael stein discovered that chevalley groups $g=g(phi,r)$ over a semilocal ring admit remarkable gauss decomposition $g=tuu^-u$, where $t=t(phi,r)$ is a split maximal torus, whereas $u=u(phi,r)$ and $u^-=u^-(phi,r)$ are unipotent radicals of two opposite borel subgroups $b=b(phi,r)$ and $b^-=b^-(phi,r)$ contai...

Weakly Closed Unipotent Subgroups in Chevalley Groups

The aim of this note is to classify all weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that 2 dimX + dim cg(X) < dim g for X a non-trivial unipotent subgroup of the connected simple algebraic group G. This shows the failure of the analogue of the so called “2F-condition” for finite groups for the adjoi...

Chevalley Groups over Commutative Rings