Pursuit curve: Difference between revisions
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{{Lie groups|Homogeneous spaces}} | |||
In the theory of [[algebraic groups]], a '''Borel subgroup''' of an [[algebraic group]] ''G'' is a maximal [[Zariski topology|Zariski closed and connected]] [[solvable group|solvable]] [[algebraic subgroup]]. | |||
For example, in the group ''GL<sub>n</sub>'' (''n x n'' invertible matrices), | |||
the subgroup of invertible [[upper triangular matrix|upper triangular matrices]] is a Borel subgroup. | |||
For groups realized over [[algebraically closed field]]s, | |||
there is a single [[conjugacy class]] of Borel subgroups. | |||
Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, [[reductive group|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. | |||
==Parabolic 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 when the homogeneous space G/B is a complete variety which is "as large as possible". | |||
For a simple algebraic group ''G'', the set of [[conjugacy class]]es of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding [[Dynkin diagram]]; the Borel subgroup corresponds to the empty set and ''G'' itself corresponding to the set of all nodes. (In general each node of the Dynkin diagram determines a simple negative root and thus a one dimensional 'root group' of ''G''---a subset of the nodes thus yields a parabolic subgroup, generated by ''B'' and the corresponding negative root groups. Moreover any parabolic subgroup is conjugate to such a parabolic subgroup.) | |||
==Lie algebra== | |||
For the special case of a [[Lie algebra]] <math>\mathfrak{g}</math> with a [[Cartan subalgebra]] <math>\mathfrak{h}</math>, given an [[order theory|ordering]] of <math>\mathfrak{h}</math>, the Borel subalgebra is the direct sum of <math>\mathfrak{h}</math> and the [[weight space]]s of <math>\mathfrak{g}</math> with positive weight. A Lie subalgebra of <math>\mathfrak{g}</math> containing a Borel subalgebra is called a [[parabolic Lie algebra]]. | |||
==See also== | |||
* [[Hyperbolic group]] | |||
==References== | |||
*{{cite conference | author=Gary Seitz | title= Algebraic Groups | booktitle=Finite and Locally Finite Groups| year=1991 | pages=45–70| editor=B. Hartley et al.}} | |||
*{{cite book | author=J. Humphreys | title=Linear Algebraic Groups | location=New York | publisher=Springer | year=1972 | isbn=0-387-90108-6}} | |||
*{{cite book | author=A. Borel | title=Essays in the History of Lie Groups and Algebraic Groups | location=Providence RI | publisher=AMS | year=2001 | isbn=0-8218-0288-7}} | |||
==External links== | |||
*{{SpringerEOM| title=Parabolic subgroup | id=Parabolic_subgroup | oldid=16195 | first=V.L. | last=Popov | authorlink = Vladimir L. Popov }} | |||
*{{SpringerEOM| title=Borel subgroup | id=Borel_subgroup | oldid=14476 | first=V.P. | last=Platonov }} | |||
[[Category:Algebraic groups]] | |||
Latest revision as of 18:01, 11 April 2013
Template:Lie groups In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the group GLn (n x n invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup.
For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups.
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.
Parabolic 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 when the homogeneous space G/B is a complete variety which is "as large as possible".
For a simple algebraic group G, the set of conjugacy classes of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding Dynkin diagram; the Borel subgroup corresponds to the empty set and G itself corresponding to the set of all nodes. (In general each node of the Dynkin diagram determines a simple negative root and thus a one dimensional 'root group' of G---a subset of the nodes thus yields a parabolic subgroup, generated by B and the corresponding negative root groups. Moreover any parabolic subgroup is conjugate to such a parabolic subgroup.)
Lie algebra
For the special case of a Lie algebra with a Cartan subalgebra , given an ordering of , the Borel subalgebra is the direct sum of and the weight spaces of with positive weight. A Lie subalgebra of containing a Borel subalgebra is called a parabolic Lie algebra.
See also
References
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