# Borel subgroup

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A maximal connected solvable algebraic subgroup of a linear algebraic group . Thus, for instance, the subgroup of all non-singular upper-triangular matrices is a Borel subgroup in the general linear group . A. Borel  was the first to carry out a systematic study of maximal connected solvable subgroups of algebraic groups. Borel subgroups can be characterized as minimal parabolic subgroups, i.e. algebraic subgroups of the group for which the quotient variety is projective. All Borel subgroups of are conjugate and, if the Borel subgroups , and the group are defined over a field , and are conjugate by an element of . The intersection of any two Borel subgroups of a group contains a maximal torus of ; if this intersection is a maximal torus, such Borel subgroups are said to be opposite. Opposite Borel subgroups exist in if and only if is a reductive group. If is connected, it is the union of all its Borel subgroups, and any parabolic subgroup coincides with its normalizer in . In such a case a Borel subgroup is maximal among all (and not only algebraic and connected) solvable subgroups of . Nevertheless, maximal solvable subgroups in which are not Borel subgroups usually exist. The commutator subgroup of a Borel subgroup coincides with its unipotent part , while the normalizer of in coincides with . If the characteristic of the ground field is 0, and is the Lie algebra of , then the subalgebra of which is the Lie algebra of the Borel subgroup of is often referred to as a Borel subalgebra in . The Borel subalgebras in are its maximal solvable subalgebras. If is defined over an arbitrary field , the parabolic subgroups which are defined over and are minimal for this property, play a role in the theory of algebraic groups over similar to that of the Borel groups. For example, two such parabolic subgroups are conjugate by an element of .