Borel subgroup

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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 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".

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.

[edit] 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 subspaces of \mathfrak{g} with positive weight.

[edit] References

  • Gary Seitz (1991). "Algebraic Groups". B. Hartley et al Finite and Locally Finite Groups: 45-70. 
  • 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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