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.

Related Topics:
Algebraic groups - Algebraic group - Zariski closed and connected - Solvable - Algebraic subgroup

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For example, in the group GLn (n x n invertible matrices),

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the subgroup of upper triangular matrices is a Borel subgroup.

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For groups realized over algebraically closed fields,

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there is a single conjugacy class of Borel subgroups.

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Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups.

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Parabolic subgroups P are also characterized, among algebraic subgroups, by the condition that G/P is a complete variety.

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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".

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Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive)

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algebraic groups, in Jacques_Tits' theory of groups with a (B,N) pair.

Related Topics:
Jacques_Tits - (B,N) pair

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Here the group B is a Borel subgroup and N is the normalizer of a maximal torus contained in B.

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The notion was introduced by Armand Borel, who played a leading role in the development of the theory of algebraic groups.

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Strictly speaking an algebraic group is a functor and a Borel subgroup is another such functor. For example in the case of

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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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