| Borel Subgroup |
Article Index for Borel |
Information AboutBorel Subgroup |
| CATEGORIES ABOUT BOREL SUBGROUP | |
| algebraic groups | |
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For example, in the group ''GLn'' (''n x n'' invertible matrices), the subgroup of Upper Triangular Matrices is a Borel subgroup. For groups realized over Algebraically Closed Field s, there is a single Conjugacy Class of Borel subgroups. Subgroups between a Borel subgroup ''B'' and the ambient group ''G'' are called parabolic subgroups. Parabolic subgroups ''P'' are also characterized, among algebraic subgroups, by the condition that ''G''/''P'' is a Complete Variety . Working over algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense. Thus ''B'' is a Borel subgroup precisely when ''G''/''B'' is a Homogeneous Space for ''G'' and a complete variety, which is "as large as possible". Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive) algebraic groups, in Jacques_Tits ' theory of groups with a (B,N) Pair . Here the group ''B'' is a Borel subgroup and ''N'' is the normalizer of a maximal torus contained in ''B''. The notion was introduced by Armand Borel , who played a leading role in the development of the theory of algebraic groups. Strictly speaking an algebraic group is a functor and a Borel subgroup is another such functor. For example in the case of ''GLn'' we did not specify the field (or commutative ring) of coefficients, so we actually have a variable family of groups. REFERENCES |
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