Borel subgroup

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 GL_n (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.

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".

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.)

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.

Lie algebra

For the special case of a Lie algebra ${\mathfrak {g}}$ with a Cartan subalgebra ${\mathfrak {h}}$ , given an ordering of ${\mathfrak {h}}$ , the Borel subalgebra is the direct sum of ${\mathfrak {h}}$ and the weight spaces of ${\mathfrak {g}}$ with positive weight. A Lie subalgebra of ${\mathfrak {g}}$ containing a Borel subalgebra is called a parabolic Lie algebra.

References

Gary Seitz (1991). "Algebraic Groups". In B. Hartley; et al. (eds.). Finite and Locally Finite Groups. pp. 45–70. {{cite conference}}: Explicit use of et al. in: |editor= (help); Unknown parameter |booktitle= ignored (|book-title= suggested) (help)
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.

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