“Abstract” homomorphisms of split Kac-Moody groups

2009 ◽  
Vol 198 (924) ◽  
pp. 0-0 ◽  
Author(s):  
Pierre-Emmanuel Caprace
Keyword(s):  
Abstract Homomorphisms

scholarly journals Morphisms between Cremona groups, and characterization of rational varieties

2014 ◽  
Vol 150 (7) ◽  
pp. 1107-1124 ◽  
Author(s):  
Serge Cantat
Keyword(s):  
Projective Variety ◽  
Abstract Group ◽  
Birational Transformations ◽  
Rational Varieties ◽  
Complex Projective Variety ◽  
Abstract Homomorphisms ◽  
Complex Projective

AbstractWe classify all (abstract) homomorphisms from the group$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\sf PGL}_{r+1}(\mathbf{C})$to the group${\sf Bir}(M)$of birational transformations of a complex projective variety$M$, provided that$r\geq \dim _\mathbf{C}(M)$. As a byproduct, we show that: (i)${\sf Bir}(\mathbb{P}^n_\mathbf{C})$is isomorphic, as an abstract group, to${\sf Bir}(\mathbb{P}^m_\mathbf{C})$if and only if$n=m$; and (ii)$M$is rational if and only if${\sf PGL}_{\dim (M)+1}(\mathbf{C})$embeds as a subgroup of${\sf Bir}(M)$.

Abstract homomorphisms of algebraic groups: problems and bibliography

1981 ◽  
Vol 9 (1) ◽  
pp. 95-114 ◽  
Author(s):  
D. James ◽  
W. Waterhouse ◽  
B. Weisfeiler
Keyword(s):  
Algebraic Groups ◽  
Abstract Homomorphisms

scholarly journals ON IMAGES OF REAL REPRESENTATIONS OF SPECIAL LINEAR GROUPS OVER COMPLETE DISCRETE VALUATION RINGS

2015 ◽  
Vol 58 (1) ◽  
pp. 263-272
Author(s):  
TALIA FERNÓS ◽  
POOJA SINGLA
Keyword(s):  
Linear Group ◽  
Positive Characteristic ◽  
Residue Field ◽  
Linear Groups ◽  
Special Linear ◽  
Special Linear Groups ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Abstract Homomorphisms ◽  
Finite Index Subgroup

AbstractIn this paper, we investigate the abstract homomorphisms of the special linear group SLn($\mathfrak{O}$) over complete discrete valuation rings with finite residue field into the general linear group GLm($\mathbb{R}$) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn($\mathfrak{O}$). For $\mathfrak{O}$ with positive characteristic, this result holds for all m ∈ ${\mathbb N}$.

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