In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski topology solvable group algebraic subgroup. For example, in the group GLn ( n x n invertible matrices), the subgroup of upper triangular matrix 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.

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