Borel and parabolic subalgebra of g lieg and have analogous properties. For example, in the general linear group gl n n x n invertible matrices, the subgroup of invertible upper triangular matrices is a borel subgroup. On conjugacy classes of closed subgroups and stabilizers of. Borel bo was the first to carry out a systematic study of maximal connected solvable subgroups of algebraic groups. By rst part of theorem above, it su ces to consider the case of borel subgroup p. A borel subgroup is a maximal solvable subgroup of g, and any such subgroup is conjugate to the subgroup b generated by t and the root groups expce. G of h into a higher rank lie group g, and deforming. G x m a h is a borel cocycle if, for all g, h g, we have agh, m ag, hmah, m for a. We can rename that borel subgroup as g, so now gis solvable. From the fact that maximal tori in b are conjugate, it follows that two maximal tori of g are conjugate.
Then, either 1g contains sl2f 2g is a borel subgroup 3g is the normalizer of a cartan. R lie g r and k r lie k r are the corresponding lie algebras and g rk r is regarded as an abelian group with the usual addition of vectors. The modality of a borel subgroup in a simple algebraic group. M x g a h is a borel cocycle if, for each g, i g, we have ax, gh ax, g axg, h for a. If moreover gacts, then xmust contain a gorbit isomorphic to gp, where p b. Let us take a cartan subgroup t of bo and let uq be the unipotent part of bo. With these few abstract notions it is hard to tell what step to take next. T kills all unipotent elements of gk, so such elements lie in uk. A subgroup pcontaining bis called a parabolic subgroup, about which more anon. On the group level, we have the parabolic subgroup q. A maximal connected solvable subgroup of g is called a borel subgroup.
Thus, for a general smooth connected a ne group gover k k, we may apply this to gr ug to get a pair of borel subgroups b and b0containing t such that r ub \r ub0 r ug schemetheoretically. The group is an example of a unipotent linear algebraic group, the group is an example of a solvable algebraic group called the borel subgroup of. Borelweil theory for groups over commutative banach algebras. It is a semidirect product of a normal unipotent group and a parabolic subgroup. Lake superior leipzig least common multiple louis st. Cocycle superrigidity for ergodic actions of nonsemisimple lie groups dave witte abstract suppose l is a semisimple levi subgroup of a connected lie group g, x is a borel gspace with finite invariant measure, and a. In the theory of algebraic groups, a borel subgroup of an algebraic group g is a maximal zariski closed and connected solvable algebraic subgroup. A pair of groups g, h is called a borel pair if g is a m. In characteristic 0 this is usually first done on the level of lie algebras. We call band any of its conjugates in ga borel subgroup.
The borel subgroup is intimately related to the notion of a parabolic subgroup. A lattice y in a connected, semisimple, real lie group g is irreducible if nt is dense in g, for every closed, connected, noncompact, normal subgroup n of g. In g glnk, a borel subgroup b is given by the upper triangular matrices. According to cartans theorem, a closed subgroup of g \displaystyle g admits a unique smooth structure which makes it an embedded lie subgroup of g \displaystyle g i. Pdf actions of complex lie groups and the borelweil.
The rst task is to control all possibilities for u a by proving that if h gis a tnormalized smooth connected k subgroup for which lie h g. It admits an action of galkq permuting the generators. Supergaussian, abelian, righttaylor functors and integral lie theory p. This is an analog of the conjugacy of cartan subgroups of a compact lie group. Those are the borel subgroups of the form b b man, where nis the complexi cation of n 0, ais the complexi cation of a 0, m z ka is the complexi cation of m 0, and b m is a borel subgroup of m.
The connected component of the identity for the usual. G0p is surjective, g0p is complete, so p is parabolic, and so pcontains a borel subgroup of g0. It is not true that for a connected algebraic r group, the lie group ga. Actions of borel subgroups on homogeneous spaces of. The weyl group w of g over q is a coxeter group with generators so. Consider the unique codimension1 torus t a kera0 red in tkilled by the nontrivial character aof t. Poisson lie group, tangent lie bialgebra of a poisson lie group, exponential map for lie bialgebras. Then by x g, x may be regarded as the set of borel subgroups of g. Let h be any p local generalized cohomology theory. Sambarino asked whether any torsionfree borel anosov subgroup of sld. Let t be a maximal torus with lie algebra h as before, and x a borel subgroup bsuch that lie b b contains h. Let gbe a compact lie group and xbe any g cwcomplex, or let gbe a discrete group and xa g cwcomplex with the property that the isotropy group of each cell is nite. If g is simply connected, then g0is a lie subgroup. Q is a real form l0 of l and that l0 is the centralizer in k0 of a circle subgroup.
The e represents the identity element of the weyl group. Bounds for the anticanonical bundle of a homogeneous. Discrete subgroups of lie groups and discrete transformation. Suppose a lie group g acts measurably on a borel space m, with a quasiinvariant measure p, and h is a second countable locally compact group.
By inspecting the dimensions of the lie algebras it. The corresponding weyl group w t is the coxeter group generated by the simple re ections fs. Since k k, so gis ksplit, we have g uo t for a smooth connected unipotent u and a torus t. In particular, a borel subgroup need not be connected with respect to other topologies, such as the topology arising from analytic structures under the natural interpretation of the group as a lie group when working over the field of real numbers, field of complex numbers, or field of padic numbers. Ergodic actions of semisimple lie groups on compact principal.
In 16, the authors address the uncountability of isomorphisms under the additional assumption that u g. We use ifflt to denote the group of c diffeomorphisms of t, and iffjt to denote the subgroup of orientation preserving diffeomorphisms. Ergodic actions of semisimple lie groups on compact. If gis a complex lie subgroup of glv, then we say that gcan be.
A lie subgroup of a lie group is a lie group that is a subset of and such that the inclusion map from to is an injective immersion and group homomorphism. The lie ko1chin theorem implies that b is a borel subgroup of. It is a consequence of the lie kolchin theorem that any connected solvable subgroup of g l n \displaystyle \mathrm gl n is conjugated into b \displaystyle b. A subgroup of a lie group is a lie subgroup if it is a submanifold. Diagonalizable, connected solvable and semisimple groups. Let gbe a complex semisimple lie group with lie algebra g, and b be a borel subgroup with lie algebra b. In particular, if g is a semisimple lie group which is connected and has no compact factors, and t is a lattice subgroup, then g, t is a borel pair. Cartan subalgebra t of a semisimple lie algebra, the root system. Let g be a simple algebraic group over an algebraically closed. G is a subgroup h 6 g so that for any subgroup k with h. H unique up to conjugation by norm gh the atlas point of view.
Hence ga is a closed lie subalgebra of a lie algebra of a linear lie group, and therefore integrates to a lie group mais62. B is equipped with the data of a root system t depending on t, and a set of simple roots tb which depends also on the choice of b. Assume l has finite center, and that the real rank of every simple factor of l is at least. By inspecting the dimensions of the lie algebras it follows that dimkl dimgb. A lie algebra is a kvector space l together with a bilinear map. Borel subgroupsnew hopf algebras from olda duality property of qthaquantized enveloping algebras in which u qu is not a hopf algebra let b tu be the standard borel subgroup of a complex lie group g.
Complete varieties, parabolic subgroups and borel subgroups. Let a solvable group bact on a nonempty complete variety x. In particular, a lie subgroup of g is parabolic if and only if it contains a borel subgroup. May 10, 2012 higher rank simple lie groups theorem stuckzimmer, 1994 if g is a simple lie group of real rank 2 and k lie groups in the theory of algebraic groups, a borel subgroup of an algebraic group g is a maximal zariski closed and connected solvable algebraic subgroup. That group q is a closed connected complex subgroup of g, and q is its own normalizer in g. We show that every balmost commutative morphism is regular, conditionally geometric, naturally pascal and pairwise brahmagupta. The weyl group for type d n is the subgroup of h n containing elements. Let p be a parabolic subgroup of g, that is, a subgroup containing a borel subgroup.
If his a parabolic subgroup, a borel subgroup a maximal torus or a maximal unipotent subgroup, then so if. For example, in the group gl n n x n invertible matrices, the subgroup of invertible upper triangular matrices is a borel subgroup. It is a surprising analogy since the algebraic structures of the groups g r and g r. For each simple lie algebra, there exists at least one associated algebraic group. The abelian group x can be viewed as the character group over k of a levi factor to of a borel subgroup b0 c g0 over q. Lie algebras corresponding to borel and parabolic subgroup of lie group g are called resp. The additive group rn has left and right invariant haar measure, namely the lebesgue measure d dx 1dx 2 dx n. Pdf the cascade of orthogonal roots and the coadjoint. Let us take xq x and let bo be the corresponding borel subgroup of g. Here t is the standard maximal torus of b and u is the maximal unipotent subgroup. Here g is a reductive lie group and h a closed subgroup. Marguliss normal subgroup theorem a short introduction.
Gcontains an elementary abelian p subgroup eof rank k, i. Request pdf actions of borel subgroups on homogeneous spaces of reductive complex lie groups and integrability let g be a real reductive lie group, k its compact subgroup. In particular, a compact homogeneous space for gmust be gisomorphic to some gp. This lie algebra integrates to the lie group glna of all invertible matrices with entries in a. By iwasawa borel subgroup of gwe mean a borel subgroup b. Two borel subgroups of go contain the rsplit cartan subgroup do, but they are conjugate and thus correspond to the same goorbit in the flag variety.
Quazitriangular lie bialgebras, factorizable lie bialgebras. Kostant, the cascade of orthogonal roots and the coadjoint structure of the nilradical of a borel subgroup of a semisimple lie group, moscow math. Despite the preceding example, borel s theorem does have interesting applications to linear algebraic groups over general ground elds, as well as useful applications even over an algebraically closed eld. A subgroup p of a group g is called a parabolic subgroup if it properly contains a borel subgroup b of g. The weyl group for type b n and c n is the hyperoctahedral group h n cn2o s n where s n acts on cn 2 by permutation. Localization and standard modules for real semisimple lie. S1 s1 r 2z 2, and pick a line r in r 2 of irrational slope. Request pdf on conjugacy classes of closed subgroups and stabilizers of borel actions of lie groups let g be a connected lie group and \mathcalcg be the space of all closed subgroups of g.
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