Borel subgroup

2020 Mathematics Subject Classification: Primary: 20G [MSN][ZBL]

A maximal connected solvable algebraic subgroup of a linear algebraic group $G$. Thus, for instance, the subgroup of all non-singular upper-triangular matrices is a Borel subgroup in the general linear group $\textrm{GL}(n)$. A. Borel [Bo] 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 $H$ of the group $G$ for which the quotient variety $G/H$ is projective. All Borel subgroups of $G$ are conjugate and, if the Borel subgroups $B_1$, $B_2$ and the group $G$ are defined over a field $k$, $B_1$ and $B_2$ are conjugate by an element of $G(k)$. The intersection of any two Borel subgroups of a group $G$ contains a maximal torus of $G$; if this intersection is a maximal torus, such Borel subgroups are said to be opposite. Opposite Borel subgroups exist in $G$ if and only if $G$ is a reductive group. If $G$ is connected, it is the union of all its Borel subgroups, and any parabolic subgroup coincides with its normalizer in $G$. In such a case a Borel subgroup is maximal among all (and not only algebraic and connected) solvable subgroups of $G$. Nevertheless, maximal solvable subgroups in $G$ which are not Borel subgroups usually exist. The commutator subgroup of a Borel subgroup $B$ coincides with its unipotent part $B_u$, while the normalizer of $B_u$ in $G$ coincides with $B$. If the characteristic of the ground field is 0, and $\def\fg{\mathfrak{g}}\fg$ is the Lie algebra of $G$, then the subalgebra $\mathfrak{b}$ of $\fg$ which is the Lie algebra of the Borel subgroup $B$ of $G$ is often referred to as a Borel subalgebra in $\fg$. The Borel subalgebras in $\fg$ are its maximal solvable subalgebras. If $G$ is defined over an arbitrary field $k$, the parabolic subgroups which are defined over $k$ and are minimal for this property, play a role in the theory of algebraic groups over $k$ similar to that of the Borel groups. For example, two such parabolic subgroups are conjugate by an element of $G(k)$ [BoTi].

References

[Bo]	A. Borel, "Groupes linéaires algébriques" Ann. of Math. (2), 64 : 1 (1956) pp. 20–82 MR0093006 Zbl 0070.26104
[BoTi]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402