On the conjecture of Borel and Tits for abstract homomorphisms of algebraic groups

The conjecture of Borel-Tits (1973) states that if $G$ and
$G’$ are algebraic groups defined over infinite fields $k$ and $k’$,
respectively, with $G$ semisimple and simply connected, then given any
abstract representation $\rho \colon G(k) \to G’(k’)$ with
Zariski-dense image, there exists a commutative finite-dimensional
$k’$-algebra $B$ and a ring homomorphism $f \colon k \to B$ such that
$\rho$ can essentially be written as a composition $\sigma \circ F$,
where $F \colon G(k) \to G(B)$ is the homomorphism induced by $f$ and
$\sigma \colon G(B) \to G’(k’)$ is a morphism of algebraic groups. We
prove this conjecture in the case that $G$ is either a universal
Chevalley group of rank $\geq 2$ or the group $\mathbf{SL}_{n, D}$,
where $D$ is a finite-dimensional central division algebra over a
field of characteristic 0 and $n \geq 3$, and $k’$ is an algebraically
closed field of characteristic 0. In fact, we show, more generally
that if $R$ is a commutative ring and $G$ is a universal
Chevalley-Demazure group scheme of rank $ \geq 2$, then abstract
representations over algebraically closed field of characteristic 0 of
the elementary subgroup $E(R) \subset G(R)$ have the expected
description. We also give applications to deformations of
representations of $E(R).$