Semisimple element of a lie group pdf

Algebraic dmodules and representation theory of semisimple. A problem on semisimple element in a lie algebra and the. In particular, we classfiy all nilpotents giving rise to semisimple and regular semisimple cyclic elements. Representations of semisimple lie algebras in prime. Notes on nilpotent elements in modular lie algebras 1. In particular, every connected semisimple lie group meaning that its lie algebra is semisimple is reductive. We can view x as an element of the lie algebra g sln. For instance, in the theory of algebraic groups, a separable element of a matrix group one with distinct eigenvalues is one that generates a separable algebra. Lie algebras, algebraic groups, and lie groups james milne.

This will lead into nitedimensional representations and the weyl character formula. For example, we show that a lie group is simple if and only if its lie algebra. We will be concerned with a semisimple algebraic group g defined over a field fc of characteristic zero. Namely we ask for the adjoint action of hon g to be semisimple. Assume l has finite center, and that the real rank of every simple factor of l is at least. Since each semisimple element lies in a maximal torus and any two maximal tori of gare conjugate, any semisimple conjugacy class of gcontains an element swhich lies in our maximal torust. What is the relationship between semisimple lie algebras and. A problem on semisimple element in a lie algebra and the reductive centralizer. Under the adjoint representation of a lie group g, an element g c g induces an automorphism adg of m. The links to gt, ca, ag, and ags in the pdf file will work if the files are placed in the. In his 1963 paper 10 the more general concept of regular element comes into play in the guise of a set denoted r but without the name \regular. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Jul 29, 2008 we give a general theory of matrix elements mes of the unitary irreducible representations uirs of linear semisimple lie groups and of reductive lie groups.

The killing dual zof f, called the characteristic element of fg. Regular elements in semisimple lie algebras let gbe a connected semisimple algebraic group over c, and let g lieg be its lie algebra. To see that this is a representation, we must show it preserves the commutation relations, that is. We develop a theory of cyclic elements in semisimple lie algebras. Let h be a fixed cartan subalgebra of l which is necessarily split. The natural grading on tg restricts to a grading on sg and ag.

Centralizers of semisimple elements in finite twisted groups. Let pbe a partial ag variety, and consider the space g pof pairs of a parabolic subalgebra p 2pand an element in it. Let 2 be a locally compact group, with a closed, unimodular, cocompact subgroup h. Ii by harishchandra in an earlier paper 5 we have established a close relationship between an irreducible representation of a semisimple lie group on a banach space and the corresponding representation of its lie algebra. An element gof gis adsemisimple if adg is semisimple in glg. Let k, l be semisimple lie algebras over an algebraically closed field of characteristic 0. If g and h are simple groups of lie type of odd characteristic such that m 1 g m 1 h and m 2 g m 2 h. Five lectures on lattices in semisimple lie groups 5 b1 a12a21 0. Nilpotent orbits in semisimple lie algebras david h. If lis a finitedimensional semisimple lie algebra over cand xis a semisimple element of l i. C and the special unitary group k sun as a maximal. Root spaces and dynkin diagrams in the previous chapter the canonical commutation relations for semisimple lie algebras were elegantly expressed in terms of roots.

We next consider semisimple conjugacy classes in g. The result is extended to the real semisimple lie algebras. But is there any deeper connection underlying the common names. Adsemisimple conjugacy classes in lie groups arxiv. This theory connects together the following things, 1 meuirs of all the representation series of a noncompact lie group, 2 meuirs of compact and noncompact forms of the. The group of fcrational points of g will be denoted gk. Recall that a cartan subalgebra is a nilpotent subalgebra which is selfnormalizing. This notion was introduced by kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. However, for a semisimple lie group over c, the algebraic theory does turn out to apply thanks to chevalleys classi cation. For semisimple lie algebras this standard form has a very rigid structure whose usefulness is surpassed only by its beauty. Associated with a lie group gis the tangent space at the identity element t 1g. By a general simple argument we prov e that a nilpotent e of a simple lie algebra g is of nilpotent type if and only if its d.

Contents 1 basic definitions and examples 2 2 theorems of engel and lie 4 3 the killing form and cartans criteria 8 4 cartan subalgebras 12 5 semisimple lie algebras 15. Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple lie algebra. Request pdf semisimple cyclic elements in semisimple lie algebras this paper is a continuation of the theory of cyclic elements in semisimple lie algebras, developed by elashvili, kac and vinberg. A conjugacy class cin gis adsemisimple if a hence every. Regular elements in semisimple lie algebras let gbe a connected semisimple algebraic group over c, and let g lie g be its lie algebra. Introduction let gbe a reductive group over an algebraically closed eld k of characteristic ph, and let g be its lie algebra. Cyclic elements in semisimple lie algebras springerlink. Furthermore, suppose that there is an injective morphism of lie algebras f. On the singular elements of a semisimple lie algebra and. This is a special case of lemma 3 in oneill 44 chapter 11. Zimmers cocycle superrigidity theorem often shows that up to cohomology, and modulo a compact group these are the only examples. We will first begin with lie groups and some differential geometry.

Every automorphism of hrad h indeed, of any semisimple lie group is volumepreserving. We will discuss the classi cation of semisimple lie algebras, root systems, the weyl group, and dynkin diagrams. Xis the borel subalgebra bx which corresponds to the point x. Semisimple conjugacy classes and classes in the weyl group. Dipolarizations in semisimple lie algebras and homogeneous. Large element orders and the characteristic of lietype. Contemporary mathematics volume cocompact subgroups of.

Element ordersin lietype groups computation ofcharacteristics. For instance, it is known that every n by n complex matrix x with zero trace is unitarily similar to a matrix with zero diagonal. On the other hand, the universal cover of sl2,r is not a real reductive group, even though its lie algebra is reductive, that is, the product of a semisimple lie algebra and an abelian lie algebra. Semisimple lie algebra an overview sciencedirect topics. Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016. If h is a lie group, and h,rad h e, then h is unimodular.

Notes on nilpotent elements in modular lie algebras 1 jordan. X, h, y is semisimple and thus forms the fcrational points of an algebraic lie algebra a by corollary 7. Endg is diagonalizable over r and is called nilpotent if adx. Let g be a connected semisimple algebraic group over c, and let g lieg be its lie algebra. Controllability of linear systems on lie groups with finite semisimple center article pdf available in siam journal on control and optimization 552 january 2016 with 77 reads. Lee is a group whose elements are organized continuously and smoothly, as opposed to discrete groups, where the elements are separatedthis makes lie groups differentiable manifolds. Also, the lie group r is reductive in this sense, since it can be viewed as the identity component of gl1,r. A semisimple element s2g is regular if its centralizer z gs fx2g jx. The problem of classifying the real reductive groups largely reduces to classifying the simple lie groups. Moreover, two elements of t are conjugate in gif and only if they lie in the same worbit ont. If both a12 and a21 are equal to 0, then the product. Let g be any connected lie group having g as its lie algebra.

Here k is a maximal compact subgroup of g, a is abelian and n is nilpotent. Root space decomposition of semisimple lie algebras. Su4,c, b is a real matrix and is hence an element in so6. Introduction to lie groups and lie algebras stony brook. A complex semisimple lie algebra must contain nonzero semisimple elements. In addition, when gis a nonsimplylaced group, we show that there are three. Let gbe a connected semisimple lie group with finite center, g kanan iwasawa decomposition of gwhere kis compact, aabelian, and nnilpotent.

This is seen even more directly in the lie algebra using nondegeneracy of the killing form. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from. We will discuss the classification of semisimple lie algebras. Example the orthogonal group on is the subgroup of glnr of elements x such that. However, the correspondence between real and complex semisimple lie algebras established with the help of the complexification is not onetoone. 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. Let v be the irreducible highestweight module over l with highest weight a. A lie algebra is reductive if and only if it is the direct sum of an abelian and a semisimple lie algebra. Lemma 1 follows immediately from the fact that if the element g lying in the semisimple group g is not semisimple, then its class of conjugate elements is not closed in g. Collingwood through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple lie algebra.

The lie algebra n spanned by x is the lie algebra of a unipotent group u in g. Some exponential inequalities for semisimple lie group. This is possible because the complexification of a real semisimple lie algebra is also semisimple see 1. In mathematics, a lie algebra is semisimple if it is a direct sum of simple lie algebras, i.

Descargar nilpotent orbits in semisimple lie algebras en. A statement on a connected semisimple noncompact lie. Structure theory of semisimple lie groups stony brook mathematics. In particular, we classfiy all nilpotents giving rise. Representations of semisimple lie groups anthony w. Gf be semisimple and let t be a maximally split maximal torus. Letmbethecentralizerofaink,0thecartaninvolutionofgwhichis identity onk, putjn onandlet g, f, a, n, m, andn denote therespective lie algebras of the groups above. Lie groups are analytic manifolds with continuous group operations. Kostants treatment of complex semisimple lie algebras in 9 achieved similar results relative to what he called principal nilpotent elements of g. Introduction let g0 be a connected semisimple lie group with. For a connected real reductive group g, the quotient manifold g k of g by a maximal compact subgroup k is a symmetric space of noncompact type. Algebraic groups are algebraic varieties with continuous group operations.

Pdf controllability of linear systems on lie groups with. Cocompact subgroups of semisimple lie groups lemma 1. An element xof g is adsemisimple if adx is semisimple in glg. It will be shown that given any element x, in a simple lie algebra 2 over c there exists a y e 2 such that the lie algebra generated by x and y is 2.

The object of the present paper is to make a deeper study of the representations of the algebra. Lie groups are named after norwegian mathematician sophus lie, who laid the foundations of the theory of continuous transformation groups. In particular, where multiplication is the tensor product one notes the following. We shall show that every irreducible representation of the group at least in. Associative kalgebras are required to have an element 1. B 2 is in the orbit corresponding to the longest weyl group element w 0. In particular, semisimple lie algebras are reductive. Gv is compact, then the group v consists of semisimple in g elements. Discrete groups, symmetric spaces, and global holonomy. Then 2 is unimodular, and hh has a finite z invariant measure. Lie groups or general real lie algebras, we introduce the following notions. Nilpotent and semisimple elements jordan decomposition def. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. Associated with such an algebraic group is a lie algebra.

The fundamental result is formulated in the introduction. The elements gs and gu are called the semisimple and unipotent part of g, respectively. Semisimple cyclic elements in semisimple lie algebras. If g is a semisimple element of a reductive lie group g, and if z is the centralizer of g in g, then not.

Interestingly, there is even an abstract jordan decomposition which holds for elements in an arbitrary semisimple lie algebra g. The kernel of this morphism is a ghomogeneous vector bundle b over x. The proof for the existence of representations is an ad hoc ver. The levi decomp osition for lie algebras says that a real lie algebra is the semidirect product of a solvable lie algebra and a semisimple lie algebra, and it follows that a connected lie group is. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its infinitesimal group and which has since become known as its lie algebra.

Besides the case in which the quotient space gv is compact, the case in which. Centralizers of semisimple elements in finite twisted groups of lie type. On the singular elements of a semisimple lie algebra and the. Introduction the following fundamental theorem was formulated in 1 let g be a semisimple algebraic real lie group of rank greater than 1, let.