Bruhat Tits Building
In mathematics , Bruhat Tits buildings are a non-Archimedean variant of symmetrical spaces . They are named after François Bruhat and Jacques Tits .
Bruhat Tits building for SL (n, K)
Be one body and be a discreet review . The evaluation ring is defined by .
The Bruhat-Tits building for the special linear group is an (n-1) -dimensional simplicial complex .
Corners : Its corners (0-simplices) are the homothesis classes of lattices in . (A lattice is a module of rank , two lattices belong to the same homothetic class if for one .)
Simplizes : Corners form a -dimensional simplex if and only if they are interspersed with
be represented with an irreducible element .
In particular, the Bruhat-Tits building is from an infinite tree whose nodes have the valence , where the remainder class field to be associated is. In this case one speaks of a Bruhat Tits tree .
In general, a Bruhat-Tits building can be defined for each reductive group over a local body .
properties
The Bruhat Tits Building is a Euclidean building, and in particular a CAT (0) room . The link of each corner is a spherical Tits building and in particular a CAT (1) room .
The group acts properly discontinuously by simplicial automorphisms on its Bruhat-Tits buildings.
The Bruhat Tits building is contractible , finite-dimensional and locally finite; the latter means that each simplex is only adjacent to a finite number of simplices .
literature
- Jean-Pierre Serre : Trees (= Springer Monographs in Mathematics. ). Translated from the French original by John Stillwell . Corrected 2nd printing of the 1980 English translation. Springer, Berlin et al. 2003, ISBN 3-540-44237-5 .
- Ian G. MacDonald: Spherical functions on a group of p-adic type (= Publications of the Ramanujan Institute. 2, ISSN 0304-9965 ). University of Madras - Ramanujan Institute, Madras 1971.
Web links
- Witte Morris: Introduction to Bruhat-Tits buildings
- Rabinoff: The Bruhat-Tits building of a p-adic Chevalley group and an application to representation theory
- Remy-Thuillier-Werner: Bruhat-Tits buildings and analytic geometry
Individual evidence
- ↑ Section 3.2 in Remy-Thuillier-Werner, op. Cit.