Algebra & Number Theory

On abstract representations of the groups of rational points of algebraic groups and their deformations

Igor Rapinchuk

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In this paper, we continue our study, begun in an earlier paper, of abstract representations of elementary subgroups of Chevalley groups of rank 2. First, we extend the methods to analyze representations of elementary groups over arbitrary associative rings and, as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form SLn,D, where D is a finite-dimensional central division algebra over a field of characteristic 0. Second, we apply the previous results to study deformations of representations of elementary subgroups of universal Chevalley groups of rank 2 over finitely generated commutative rings.

Article information

Algebra Number Theory, Volume 7, Number 7 (2013), 1685-1723.

Received: 9 June 2012
Revised: 15 June 2012
Accepted: 7 September 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20G15: Linear algebraic groups over arbitrary fields
Secondary: 14L15: Group schemes

abstract homomorphisms algebraic groups rigidity character varieties


Rapinchuk, Igor. On abstract representations of the groups of rational points of algebraic groups and their deformations. Algebra Number Theory 7 (2013), no. 7, 1685--1723. doi:10.2140/ant.2013.7.1685.

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