9,051 research outputs found
Irreducible subgroups of simple algebraic groups - a survey
Let be a simple linear algebraic group over an algebraically closed field
of characteristic , let be a proper closed subgroup of
and let be a nontrivial finite dimensional irreducible rational
-module. We say that is an irreducible triple if is
irreducible as a -module. Determining these triples is a fundamental
problem in the representation theory of algebraic groups, which arises
naturally in the study of the subgroup structure of classical groups. In the
1980s, Seitz and Testerman extended earlier work of Dynkin on connected
subgroups in characteristic zero to all algebraically closed fields. In this
article we will survey recent advances towards a classification of irreducible
triples for all positive dimensional subgroups of simple algebraic groups.Comment: 31 pages; to appear in the Proceedings of Groups St Andrews 201
Holography principle and arithmetic of algebraic curves
According to the holography principle (due to G.`t Hooft, L. Susskind, J.
Maldacena, et al.), quantum gravity and string theory on certain manifolds with
boundary can be studied in terms of a conformal field theory on the boundary.
Only a few mathematically exact results corroborating this exciting program are
known. In this paper we interpret from this perspective several constructions
which arose initially in the arithmetic geometry of algebraic curves. We show
that the relation between hyperbolic geometry and Arakelov geometry at
arithmetic infinity involves exactly the same geometric data as the Euclidean
AdS_3 holography of black holes. Moreover, in the case of Euclidean AdS_2
holography, we present some results on bulk/boundary correspondence where the
boundary is a non-commutative space.Comment: AMSTeX 30 pages, 7 figure
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