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 .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ mathfrak {g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
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 .
![{\ mathfrak {a}} \ subset {\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4125c3d45d6304dac376e53ab2a6529969589d)
![{\ mathfrak {a}} ^ {+} \ subset {\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc58b1694ca5a39c66ed5a7c0bd6a627825f861)
![R ^ {+} \ subset R](https://wikimedia.org/api/rest_v1/media/math/render/svg/e957b408014d92ae48b844c59e9d0f328819d72a)
![\ Delta \ subset R ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab017486c3924158d9849078fb8d344a01f304e)
Minimal parabolic subgroup
The minimal parabolic subgroup to be associated is the sub-lie group
![{\ mathfrak a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b)
![P \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8fef3769b7a5d882c6051dfa15935a4fc80e8f)
with Lie algebra
-
,
wherein the centralizer of and the root area of the positive root respectively.
![{\ mathfrak {z}} ({\ mathfrak {a}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/5737ee645e46c01b9313c1389959b04955c3e2bc)
![{\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b)
![{\ mathfrak {g}} _ {\ alpha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ca5ac242eb190abe22a20ff71139b10d7b935a)
![\alpha](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)
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 .
![H \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/15eb570a68da46b0458ff0ead693a82467ab8ddd)
![P \ subset H](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef131f6523262c7b388e7a1669ce48b9110412e9)
Langlands decomposition
You have the decomposition
![{\ mathfrak {p}} = {\ mathfrak {n}} \ oplus {\ mathfrak {a}} \ oplus {\ mathfrak {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd16d7475ed9e00a94e5400f2bf5fbfec62de22)
With
![{\ mathfrak {n}} = \ sum _ {{\ alpha \ in R ^ {+}}} {\ mathfrak {g}} _ {\ alpha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c8d4ea716f07de8a927fd06c30f2d235dac733)
and , where the Lie algebra denotes, that is, the Lie algebra of a maximally compact group , in particular .
![{\ mathfrak {m}} = {\ mathfrak {k}} \ cap {\ mathfrak {z}} ({\ mathfrak {a}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/956871f03caf44d4aae42276a54cfd3c81fca4fb)
![{\ mathfrak {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a94eb54c7bdae2f76ad4d43f210dd71b5fa2beb4)
![{\ mathfrak {g}} = {\ mathfrak {k}} \ oplus {\ mathfrak {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/578c1225c7847477ccbafa7548bb88f3b3778fc0)
![K \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/80f658720077576ddab27442f5d32b8bcf0a7814)
![{\ mathfrak {z}} ({\ mathfrak {a}}) = {\ mathfrak {m}} \ oplus {\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9db96c9243009750a030fb3c097839a8e89e943)
The corresponding decomposition
![P = NAM](https://wikimedia.org/api/rest_v1/media/math/render/svg/e778990792db4617f41f41e68734b24e0a173ef8)
is called the Langlands decomposition of .
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
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.
![{\ mathfrak a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b)
![I \ subset \ Delta](https://wikimedia.org/api/rest_v1/media/math/render/svg/b41c09a827d0dd4cbff70d8d1f8f12ab338234be)
![\ emptyset \ subset \ Delta](https://wikimedia.org/api/rest_v1/media/math/render/svg/05c1f894f194d43174fcfc6c105ad568e4422b77)
![R ^ {I} \ subset R ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3012b60e857ef95f5113699945c9751676edb9c1)
![I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f)
![\ alpha ^ {\ vee} \ in {\ mathfrak {a}} ^ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed27717709474eb3b7bbcf5ad7140dd54a81335)
![\ alpha \ in {\ mathfrak {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d95c783444fc06161a471b2bb9e7ad1cd26d4e)
![{\ mathfrak {a}} ^ {I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/842b893787ce81c6d59d0986ae1efc20fc7d41c5)
![{\ mathfrak {a}} _ {I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/259f397e2cbf55b2e54f90cbe82df4c9bed1ca47)
We look at
![{\ displaystyle {\ mathfrak {a}} _ {I}: = \ bigcap _ {\ alpha \ in I} \ ker (\ alpha ^ {\ vee}) \ subset {\ mathfrak {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a89ad36aa843a6168b3b6299014658ace82a3d)
![{\ mathfrak {n}} _ {I}: = \ sum _ {{\ alpha \ in R ^ {+} \ setminus R ^ {I}}} {\ mathfrak {g}} _ {\ alpha} \ subset {\ mathfrak {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d01e14cd42dce2f6ee57c8cc7f7884b5ea9d1561)
![{\ displaystyle {\ mathfrak {m}} _ {I}: = {\ mathfrak {m}} \ oplus {\ mathfrak {a}} ^ {I} \ oplus \ sum _ {\ pm \ alpha \ in R ^ {I}} {\ mathfrak {g}} _ {\ alpha} \ supset {\ mathfrak {m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f006d6d814bdec1184d22391cd0248e1c53f1b)
and
-
.
is the "standard parabolic sub-algebra" of zu . Note that the standard parabolic subalgebras depend on the choice of the positive Weyl chamber .
![\ mathfrak {g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
![I \ subset R ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a2da12663a8c356bcec3625531d87d4d7da837)
![{\ mathfrak {a}} ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a68f1757049489885c167f9b3314ca0a46b67ff)
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 .
![{\ mathfrak {p}} \ subset {\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30df84830c7400071513080b9cc4763a789869da)
![{\ mathfrak {p}} _ {I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b946eb1af3f9eee44527cfa1f49815d8cc8be79a)
![{\ mathfrak {a}} ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a68f1757049489885c167f9b3314ca0a46b67ff)
![I \ subset \ Delta](https://wikimedia.org/api/rest_v1/media/math/render/svg/b41c09a827d0dd4cbff70d8d1f8f12ab338234be)
The associated parabolic subgroup of a parabolic sub-algebra is defined as the normalizer of in .
![P \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8fef3769b7a5d882c6051dfa15935a4fc80e8f)
![{\ mathfrak {p}} \ subset {\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30df84830c7400071513080b9cc4763a789869da)
![{\ mathfrak {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
For a Weyl chamber and a subset , the parabolic subgroup belonging to it is designated with . Each parabolic subgroup contains the minimum parabolic subgroup .
![{\ mathfrak {a}} ^ {+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a68f1757049489885c167f9b3314ca0a46b67ff)
![I \ subset \ Delta](https://wikimedia.org/api/rest_v1/media/math/render/svg/b41c09a827d0dd4cbff70d8d1f8f12ab338234be)
![PI}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c820b475a5ff10f9b4cbd18652123c015143ace8)
![{\ mathfrak {p}} _ {I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b946eb1af3f9eee44527cfa1f49815d8cc8be79a)
![PI}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c820b475a5ff10f9b4cbd18652123c015143ace8)
![P _ {\ emptyset} \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd79aa674a85483ef3ef35e26002b6289b7bf344)
In this case, too, you have the Langlands decomposition
-
.
The term “parabolic sub-algebra” or “parabolic subgroup” goes back to Godement .
Example SL (n, R)
A Cartan sub-algebra of Lie algebra
![{\ mathfrak {sl}} (n, \ mathbb {R}) = \ left \ {A \ in Mat (n, \ mathbb {R}) \ colon \ operatorname {trace} (A) = 0 \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1822613f575e17191d99f6b7ef1d5cc620366fb8)
is
-
.
As a positive Weyl Chamber you can
![{\ mathfrak {a}} ^ {+} = \ left \ {(\ operatorname {diag} (t_ {1}, \ ldots, t_ {n}) \ in {\ mathfrak {a}} \ colon t_ {1 }> t_ {2}> \ ldots> t_ {n} \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb97c0df49d48e3633ae240f1e3c969e5eeb48d4)
choose. Then the Lie algebra of the upper triangular matrices with -en on the diagonal and .
![{\ mathfrak {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fab30f69b7fab337592fdb8b5384bf004f88c574)
![{\ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\ mathfrak {m}} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9abe29dbc2a7e481fdfdd420192a94f7c4f1c9)
The Langlands decomposition of is
![P _ {\ emptyset}](https://wikimedia.org/api/rest_v1/media/math/render/svg/258192995f98fbe20b2bceba093bd3c002e428b5)
![P _ {\ emptyset} = MAN](https://wikimedia.org/api/rest_v1/media/math/render/svg/b369d8c7a0ec26db7230fdbebff23378048e754e)
With
-
,
-
,
-
the group of the upper triangular matrices with -en on the diagonal.![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
The Borel group is therefore the group of the upper triangular matrices, every other Borel group is too conjugated.
![P _ {\ emptyset}](https://wikimedia.org/api/rest_v1/media/math/render/svg/258192995f98fbe20b2bceba093bd3c002e428b5)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
The maximum standard parabolic subsets, i.e. H. those for which there is only one element
![\ Delta \ setminus I](https://wikimedia.org/api/rest_v1/media/math/render/svg/aea6c2049b8b3f0b6a5a59b3149694afb0aadf56)
![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 nk}}, D \ in M _ {{nk \ times nk}} \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b08839486a6010b27c74ae9e37a05e3b7e2fd4c6)
for .
![k = 1, \ ldots, n-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/db3a592f61b9c1ecd2cbf868095d127bf1caa130)
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 .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G / P](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4b9d759f4ca0add231cb2cd7a8d475cb35374e)
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.
![B \ subset G](https://wikimedia.org/api/rest_v1/media/math/render/svg/be4aa2b7097e39aa99228d4a09765f94d33440ad)
![k = \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf4d723f5664de728ebf62b66da9f83cb55d4fe)
![{\ displaystyle k = \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb48f4addc7d427950351c63d4b5918e8ba61617)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7255d995e0278893df2f5ad7235b262e796b1661)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![G / B](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b415d3faa81ad94d1a2d5c540fbcf48e6f8ba16)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d6224361d2f21c9b24d89474103e921034cf8f5)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
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.
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![N](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![{\ displaystyle W: = N / H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8243031e454f7d6fbfb0fd7925e46bf6b4c4be65)
![{\ displaystyle (G, B, N, S)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/681fa8c70d9004834c2874ebb9383335e9505db1)
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
Individual evidence
-
^ 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)
-
^ BH Bowditch: Discrete parabolic groups. In: J. Differential Geom. 38 (1993) no. 3, pp. 559-583.