Parabolic subgroup

In mathematics , the concept of parabolic subgroups is an important term from the theory of algebraic groups and, more generally, the theory of Lie groups . Minimal parabolic groups are called Borel groups . A classic example of a (minimal) parabolic group is the group of invertible upper triangular matrices as a subgroup of the general linear group .

Another, non-equivalent, use of the term "parabolic subgroup" is found in the theory of Klein groups or the theory of convergence groups: here a parabolic subgroup is a group whose elements are parabolic isometries with the same fixed point.

Lie groups

Let it be a Lie group and its Lie algebra . ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$

Let be a Cartan sub-algebra and its root system . Choose a Weyl chamber and designate the corresponding positive roots with . It's the simple roots . ${\displaystyle {\mathfrak {a}}\subset {\mathfrak {g}}}$ ${\displaystyle ({\mathfrak {a}},R)}$ ${\displaystyle {\mathfrak {a}}^{+}\subset {\mathfrak {a}}}$ ${\displaystyle R^{+}\subset R}$ ${\displaystyle \Delta \subset R^{+}}$

Minimal parabolic subgroup

The minimal parabolic subgroup to be associated is the sub-lie group ${\displaystyle {\mathfrak {a}}}$

${\displaystyle P\subset G}$

with Lie algebra

${\displaystyle {\mathfrak {p}}={\mathfrak {z}}({\mathfrak {a}})\oplus \sum _{\alpha \in R^{+}}{\mathfrak {g}}_{\alpha }}$,

wherein the centralizer of and the root area of the positive root respectively. ${\displaystyle {\mathfrak {z}}({\mathfrak {a}})}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {g}}_{\alpha }}$ ${\displaystyle \alpha }$

The minimal parabolic subsets are also referred to as Borel subsets.

Definition of a parabolic subgroup

A subgroup is called parabolic if there is a minimal parabolic subgroup with it . ${\displaystyle H\subset G}$ ${\displaystyle P\subset H}$

Langlands decomposition

You have the decomposition

${\displaystyle {\mathfrak {p}}={\mathfrak {n}}\oplus {\mathfrak {a}}\oplus {\mathfrak {m}}}$

With

${\displaystyle {\mathfrak {n}}=\sum _{\alpha \in R^{+}}{\mathfrak {g}}_{\alpha }}$

and , where the Lie algebra denotes, that is, the Lie algebra of a maximally compact group , in particular . ${\displaystyle {\mathfrak {m}}={\mathfrak {k}}\cap {\mathfrak {z}}({\mathfrak {a}})}$ ${\displaystyle {\mathfrak {k}}}$ ${\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}$ ${\displaystyle K\subset G}$ ${\displaystyle {\mathfrak {z}}({\mathfrak {a}})={\mathfrak {m}}\oplus {\mathfrak {a}}}$

The corresponding decomposition

${\displaystyle P=NAM}$

is called the Langlands decomposition of . ${\displaystyle P}$

Parabolic subgroups

The parabolic subgroups associated with a Cartan algebra correspond to the subsets (the minimal parabolic subgroup corresponds to the subset ); they are obtained with the following construction, where the linear combinations of elements in , as well as the dual of defined by the Killing form and the orthogonal complement (with regard to the killing form) designated by. ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle I\subset \Delta }$ ${\displaystyle \emptyset \subset \Delta }$ ${\displaystyle R^{I}\subset R^{+}}$ ${\displaystyle I}$ ${\displaystyle \alpha ^{\vee }\in {\mathfrak {a}}^{*}}$ ${\displaystyle \alpha \in {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}^{I}}$ ${\displaystyle {\mathfrak {a}}_{I}}$

We look at

${\displaystyle {\mathfrak {a}}_{I}:=\bigcap _{\alpha \in I}\ker(\alpha ^{\vee })\subset {\mathfrak {a}}}$

${\displaystyle {\mathfrak {n}}_{I}:=\sum _{\alpha \in R^{+}\setminus R^{I}}{\mathfrak {g}}_{\alpha }\subset {\mathfrak {n}}}$

${\displaystyle {\mathfrak {m}}_{I}:={\mathfrak {m}}\oplus {\mathfrak {a}}^{I}\oplus \sum _{\pm \alpha \in R^{I}}{\mathfrak {g}}_{\alpha }\supset {\mathfrak {m}}}$

and

${\displaystyle {\mathfrak {p}}_{I}:={\mathfrak {a}}_{I}\oplus {\mathfrak {n}}_{I}\oplus {\mathfrak {m}}_{I}}$.

${\displaystyle {\mathfrak {p}}_{I}}$is the "standard parabolic sub-algebra" of zu . Note that the standard parabolic subalgebras depend on the choice of the positive Weyl chamber . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle I\subset R^{+}}$ ${\displaystyle {\mathfrak {a}}^{+}}$

A sub-algebra is called a parabolic sub-algebra if it is conjugated to a standard parabolic sub-algebra for a Weyl chamber and a subset . ${\displaystyle {\mathfrak {p}}\subset {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {p}}_{I}}$ ${\displaystyle {\mathfrak {a}}^{+}}$ ${\displaystyle I\subset \Delta }$

The associated parabolic subgroup of a parabolic sub-algebra is defined as the normalizer of in . ${\displaystyle P\subset G}$ ${\displaystyle {\mathfrak {p}}\subset {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle G}$

For a Weyl chamber and a subset , the parabolic subgroup belonging to it is designated with . Each parabolic subgroup contains the minimum parabolic subgroup . ${\displaystyle {\mathfrak {a}}^{+}}$ ${\displaystyle I\subset \Delta }$ ${\displaystyle P_{I}}$ ${\displaystyle {\mathfrak {p}}_{I}}$ ${\displaystyle P_{I}}$ ${\displaystyle P_{\emptyset }\subset G}$

In this case, too, you have the Langlands decomposition

${\displaystyle P_{I}=M_{I}A_{I}N_{I}}$.

The term “parabolic sub-algebra” or “parabolic subgroup” goes back to Godement .

Example SL (n, R)

A Cartan sub-algebra of Lie algebra

${\displaystyle {\mathfrak {sl}}(n,\mathbb {R} )=\left\{A\in Mat(n,\mathbb {R} )\colon \operatorname {Spur} (A)=0\right\}}$

is

${\displaystyle {\mathfrak {a}}=\left\{(\operatorname {diag} (t_{1},\ldots ,t_{n})\colon t_{1}+\ldots +t_{n}=0\right\}}$.

As a positive Weyl Chamber you can

${\displaystyle {\mathfrak {a}}^{+}=\left\{(\operatorname {diag} (t_{1},\ldots ,t_{n})\in {\mathfrak {a}}\colon t_{1}>t_{2}>\ldots >t_{n}\right\}}$

choose. Then the Lie algebra of the upper triangular matrices with -en on the diagonal and . ${\displaystyle {\mathfrak {n}}}$ ${\displaystyle 0}$ ${\displaystyle {\mathfrak {m}}=0}$

The Langlands decomposition of is ${\displaystyle P_{\emptyset }}$

${\displaystyle P_{\emptyset }=MAN}$

With

${\displaystyle M=\left\{\operatorname {diag} (\pm 1,\ldots ,\pm 1)\right\}}$,

${\displaystyle A=\left\{\operatorname {diag} (a_{1},\ldots ,a_{n})\colon a_{1},\ldots ,a_{n}>0,a_{1}\ldots a_{n}=1\right\}}$,

${\displaystyle N}$the group of the upper triangular matrices with -en on the diagonal. ${\displaystyle 1}$

The Borel group is therefore the group of the upper triangular matrices, every other Borel group is too conjugated. ${\displaystyle P_{\emptyset }}$ ${\displaystyle B}$ ${\displaystyle B}$

The maximum standard parabolic subsets, i.e. H. those for which there is only one element ${\displaystyle \Delta \setminus I}$

${\displaystyle P_{k}=\left\{{\begin{pmatrix}A&B\\0&D\end{pmatrix}}\in SL(n,\mathbb {R} )\colon A\in M_{k\times k},B\in M_{k\times n-k},D\in M_{n-k\times n-k}\right\}}$

for . ${\displaystyle k=1,\ldots ,n-1}$

Algebraic groups

A parabolic subgroup of an algebraic group defined over a field is a Zariski-closed subgroup , for which the quotient is a projective variety . ${\displaystyle k}$ ${\displaystyle G}$ ${\displaystyle P\subset G}$ ${\displaystyle G/P}$

One can show that a subgroup is parabolic if and only if it contains a Borel subgroup. (A Borel subgroup is a maximally Zariski-closed, connected , resolvable , algebraic subgroup.) Borel subgroups are thus minimal parabolic groups. In the case of or , the definition is the same as that given above. ${\displaystyle P\subset G}$ ${\displaystyle B\subset G}$ ${\displaystyle k=\mathbb {R} }$ ${\displaystyle k=\mathbb {C} }$

example

A Borel subgroup of is the group of invertible upper triangular matrices. In this case the quotient is the flag variety . ${\displaystyle G=SL(n,\mathbb {C} )}$ ${\displaystyle B}$ ${\displaystyle G/B}$

Each Borel subgroup of is too conjugate. More generally, for algebraic groups over algebraically closed fields , there is exactly one conjugation class of Borel subgroups. ${\displaystyle SL(n,\mathbb {C} )}$ ${\displaystyle B}$

Tits system

→ Main article : Tits system

Let be a reductive algebraic group and a Borel subgroup that contains a maximal torus . Let be the normalizer of in and a minimal generating system of . Then there is a tits system. ${\displaystyle G}$ ${\displaystyle B}$ ${\displaystyle H}$ ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle S}$ ${\displaystyle W:=N/H}$ ${\displaystyle (G,B,N,S)}$

Small groups

In the context of small groups , the term "parabolic subgroup" is often used with a different meaning, namely as a group of parabolic isometries that have a common fixed point and consequently depict the horospheres around this point. This use is not equivalent to that described above.

More generally, a subgroup of a convergence group is called a parabolic subgroup if it is infinite, has a global fixed point , and does not contain any loxodromic elements .

literature

• Armand Borel, Lizhen Ji: Compactifications of symmetric and locally symmetric spaces. (= Mathematics: Theory & Applications). Birkhäuser, Boston, MA 2006, ISBN 0-8176-3247-6 .

Web links

• Parabolic subgroup (Encyclopedia of Mathematics)
• Borel subgroup (Encyclopedia of Mathematics)
• Alfred Noël: Tits Systems, parabolic subgroups, parabolic subalgebras

Individual evidence

1. ^ Armand Borel: Essays in the history of Lie groups and algebraic groups. (= History of Mathematics. 21). American Mathematical Society, Providence, RI; London Mathematical Society, Cambridge 2001, ISBN 0-8218-0288-7 (Chapter VI, Section 2)
2. ^ BH Bowditch: Discrete parabolic groups. In: J. Differential Geom. 38 (1993) no. 3, pp. 559-583.