|Group theory → 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.
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.
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.)
Let . A Borel subgroup of is the set of upper triangular matrices
and the maximal proper parabolic subgroups of containing are
Also, a maximal torus in is
It should be clear that this is isomorphic to the algebraic torus .
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.
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- J. Humphreys (1972). Linear Algebraic Groups. New York: Springer. ISBN 0-387-90108-6.
- A. Borel (2001). Essays in the History of Lie Groups and Algebraic Groups. Providence RI: AMS. ISBN 0-8218-0288-7.
- Brion, Michel. "Lectures on the geometry of flag varieties" (PDF).
- Popov, V.L. (2001) , "Parabolic subgroup", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Platonov, V.P. (2001) , "Borel subgroup", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4