Difference between pages "Riemannian metric" and "Borel subgroup"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex,msc,links) |
||
| Line 1: | Line 1: | ||
| − | + | {{MSC|20G}} | |
| + | {{TEX|done}} | ||
| − | + | A maximal connected solvable algebraic subgroup of a | |
| − | + | [[Linear algebraic group|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 | |
| − | + | {{Cite|Bo}} was the first to carry out a systematic study of maximal connected [[solvable group | solvable]] subgroups of algebraic groups. Borel subgroups can be characterized as minimal [[parabolic subgroup | 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|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 $G(k)$ | |
| − | + | {{Cite|BoTi}}. | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | The | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Bo}}||valign="top"| A. Borel, "Groupes linéaires algébriques" ''Ann. of Math. (2)'', '''64''' : 1 (1956) pp. 20–82 {{MR|0093006}} {{ZBL|0070.26104}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|BoTi}}||valign="top"| A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'', '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 17:51, 27 April 2012
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 |
Riemannian metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_metric&oldid=14477