Borel subgroup

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 when the homogeneous space G/B is a complete variety which is "as large as possible".

For a simple algebraic group G, the set of conjugacy classes of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding Dynkin diagram; the Borel subgroup corresponds to the empty set and G itself corresponding to the set of all nodes. (In general each node of the Dynkin diagram determines a simple negative root and thus a one-dimensional 'root group' of G–a subset of the nodes thus yields a parabolic subgroup, generated by B and the corresponding negative root groups. Moreover, any parabolic subgroup is conjugate to such a parabolic subgroup.)

Let ${\displaystyle G=GL_{4}(\mathbb {C} )}$. A Borel subgroup ${\displaystyle B}$ of ${\displaystyle G}$ is the set of upper triangular matrices

${\displaystyle \left\{A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&0&a_{44}\end{bmatrix}}:\det(A)\neq 0\right\}}$

and the maximal proper parabolic subgroups of ${\displaystyle G}$ containing ${\displaystyle B}$ are

${\displaystyle \left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&a_{32}&a_{33}&a_{34}\\0&a_{42}&a_{43}&a_{44}\end{bmatrix}}\right\},{\text{ }}\left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&a_{43}&a_{44}\end{bmatrix}}\right\},{\text{ }}\left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\0&0&0&a_{44}\end{bmatrix}}\right\}}$

Also, a maximal torus in ${\displaystyle B}$ is

${\displaystyle \left\{{\begin{bmatrix}a_{11}&0&0&0\\0&a_{22}&0&0\\0&0&a_{33}&0\\0&0&0&a_{44}\end{bmatrix}}:a_{11}\cdot a_{22}\cdot a_{33}\cdot a_{44}\neq 0\right\}}$

This is isomorphic to the algebraic torus ${\displaystyle (\mathbb {C} ^{*})^{4}={\text{Spec}}(\mathbb {C} [x^{\pm 1},y^{\pm 1},z^{\pm 1},w^{\pm 1}])}$.[1]

For the special case of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ with a Cartan subalgebra ${\displaystyle {\mathfrak {h}}}$, given an ordering of ${\displaystyle {\mathfrak {h}}}$, the Borel subalgebra is the direct sum of ${\displaystyle {\mathfrak {h}}}$ and the weight spaces of ${\displaystyle {\mathfrak {g}}}$ with positive weight. A Lie subalgebra of ${\displaystyle {\mathfrak {g}}}$ containing a Borel subalgebra is called a parabolic Lie algebra.

• Hyperbolic group
• Cartan subgroup
• Mirabolic subgroup

• A. Borel (2001). Essays in the History of Lie Groups and Algebraic Groups. Providence RI: AMS. ISBN 0-8218-0288-7.
• J. Humphreys (1972). Linear Algebraic Groups. New York: Springer. ISBN 0-387-90108-6.
• Milne, J. S. (2017), Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, doi:10.1017/9781316711736, ISBN 978-1107167483, MR 3729270
• Gary Seitz (1991). "Algebraic Groups". In B. Hartley; et al. (eds.). Finite and Locally Finite Groups. pp. 45–70.

Specific

• Popov, V.L. (2001) [1994], "Parabolic subgroup", Encyclopedia of Mathematics, EMS Press
• Platonov, V.P. (2001) [1994], "Borel subgroup", Encyclopedia of Mathematics, EMS Press