The oneparameter sub groups of are precisely the maximal integral curves of leftinvariant vector fields starting at the identity. The cartan theorem alluded to in the title postulates the existence of a natural. The historically incorrect naming of the cartanlie theorem as the third lie theorem is largely due to the influence of a book based on lectures of jeanpierre serre lie algebras and lie groups, w. Remark the dimension of the cartan subalgebra constructed in cartans theorem, theorem 4, equals the rank of g. Cartans theorem may refer to several mathematical results by elie cartan. 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. This theorem mostly reduces the study of arbitrary lie groups to the study of finite. Any closed subgroup hof a lie group gis a lie subgroup and thus a submanifold of g. Pdf an analogue of the hardy theorem for the cartan. Suppose g is a lie group and h a closed subgroup of g, i. The weyl group, and adinvariant inner products 259 4. Onewayto prove this theorem is the so called unitarian trick of weyl.
By theorem 2 and a standard argument involving the weyl group we also obtain the following. An embedding theorem for automorphism groups of cartan geometries uri bader. Brie y, since gc is a connected and topologically simply connected lie group with gr the xed points of the involution given by complex conjugation, the problem is reduced to showing that any invo. Lecture 1 lie groups and the maurercartan equation january 11, 20 1 lie groups a lie group is a di erentiable manifold along with a group structure so that the group operations of products and inverses are di erentiable. It is not difficult to see that 5 is equivalent to. Lie algebras are an essential tool in studying both algebraic groups and lie groups. At this stage one has an extensive supply of examplesrotation groups, for example, and many others. In mathematics, the closedsubgroup theorem sometimes referred to as cartan s theorem is a theorem in the theory of lie groups. Cartans work on infinite dimensional lie al gebras, exterior differential calculus, differential ge ometry, and, above all, the representation theory of semisimple lie algebras was of supreme value. Introduction cartan type lie algebras are lie subalgebras of algebraic vector elds on a at a ne space fn, where fis. The universal cover of a connected lie group is a lie group. Now, its a standard result that a lie subgroup of a lie group g is topologically closed in g.
Nov 18, 2008 now, its a standard result that a lie subgroup of a lie group g is topologically closed in g. Theorem b is stated in cohomological terms a formulation that cartan 1953, p. The affine group of a lie group 353 chevalleys theorem on the topology of solvable groups 2, the fact that the universal covering of sl2, r is the only simple lie group homeomorphic to euclidean space, and the global leviwhitehead decomposition of g. An element a2ginduces three standard di eomorphisms l a. Weyls theorem, which says this is true for any semisimple lie algebra g. This is known as the closed subgroup theorem or cartan s theorem. Structure theory of semisimple lie groups stony brook mathematics. Now using invariant integration one shows that every representation of a compact group is unitary. If g0 is a real semisimple lie algebra, then g0 has a cartan involution.
Schur duality for the cartan type lie algebrawn kyo nishiyama communicated by g. The lie algebra g is semisimple if and only if b is nondegenerate. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology. Lie has proved earlier just the infinitesimal version local solvability of maurercartan equations see maurer. Cartan s theorem states that the converse of this result is true at least for real lie groups. The quotient of a lie group by a closed normal subgroup is a lie group. Chapter i develops the basic theory of lie algebras, including the fundamental theorems of engel, lie, cartan, weyl, ado, and poincarebirkhoffwitt. Remark the dimension of the cartan subalgebra constructed in cartan s theorem, theorem 4, equals the rank of g. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology theorem a. A real form of gis an antiholomorphic involutive automorphism. We also show that maximal compact subgroups of the radical are contained in cartan subgroups, and for a connected solvable lie group, cartan subgroups are same as those of the centralizer of maximal compact. Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016.
Using the researches of sophus lie and wilhelm killing, cartan 9 in his 1894 thesis, completed the classification of finitedimensional simple. Let g be a connected semisimple lie group with lie algebra g. Any two cartan involutions are conjugate via inn g0. For example, the group r is the universal cover of the circle group s 1. A finitedimensional lie algebra g over r is integrable, that is, there exists a lie group g with lieg. We study the automorphism group of a cartan geometry, and prove an embedding theorem analogous to a result of zimmer for automorphism groups of gstructures. Suppose g is a compact lie group and n is a closed normal subgroup of g acting freely on a smooth manifold x. Hence the full theorem is properly called the cartanlie theorem. In dealing with the algebra of angular momentum j, it is useful to single out an operator usually j z to be diagonalised and then to reexpress the. Classification and construction of semisimple lie algebras.
Write h1g for the galois cohomology pointed set h1galcr. Then a basis can be chosen for v with respect to which we obtain a matrix representation. C are the same lie algebra, a fact of some importance in quantum mechanics. Theorem lies theorem let g be a solvable lie algebra and v a. Lie groups and lie algebras, addisonwesley 1975 translated from french comments see also lie algebra, semisimple for a description of the special case of a chevalley basis. In mathematics, cartan s theorems a and b are two results proved by henri cartan around 1951, concerning a coherent sheaf f on a stein manifold x.
Lq version of hardys theorem on nilpotent lie groups, forum math. Lies third theorem, an equivalence between lie algebras and simplyconnected lie groups. It states that if h is a closed subgroup of a lie group g, then h is an embedded lie group with the smooth structure and hence the group topology agreeing with the embedding. Since the group of di eomorphisms of a compact manifold forms a regular fr echet lie group, an application of our main result yields a theorem on integration of nitedimensional lie algebras of vector elds. The cartan decomposition of a complex semisimple lie algebra. Closedsubgroup theorem, 1930, that any closed subgroup of a lie group is a lie subgroup. Since the group of di eomorphisms of a compact manifold forms a regular fr echetlie group, an application of our main result yields a theorem on integration of nitedimensional lie algebras of vector elds. Pdf an analogue of the hardy theorem for the cartan motion. Lie theory has its name from the work of sophus lie 6, who studied certain transformation groups, that is, the groups of symmetries of algebraic or geometric objects that are now called lie groups. Lie groups, lie algebras, and their representations. Cartan and iwasawa decompositions in lie theory 5 theorem 3. An embedding theorem for automorphism groups of cartan. Theorem of the highest weight, that the irreducible representations of lie algebras or lie groups are classified by their highest weights. Classically, in a real lie group this is a closed connected subgroup whose lie algebra is a cartan subalgebra of the lie algebra of the given group nilpotent and equal to its normalizer.
The automorphisms of g form a group autg under composition of maps. In mathematics, cartans theorems a and b are two results proved by henri cartan around 1951, concerning a coherent sheaf f on a stein manifold x. Salamon eth zuric h 20 november 2019 contents 1 complex lie groups 2 2 first existence proof 5 3 second existence proof 8 4 hadamards theorem 16 5 cartans xed point theorem 18 6 cartan decomposition 20 7 matrix factorization 25. Lecture 1 lie groups and the maurercartan equation. Lecture 1 lie groups and the maurer cartan equation january 11, 20 1 lie groups a lie group is a di erentiable manifold along with a group structure so that the group operations of products and inverses are di erentiable. Introduction to lie groups and lie algebras stony brook. In dealing with the algebra of angular momentum j, it is useful to single out an operator. Let g be a connected simply connected nilpotent lie group.
Cartanweyl basis from this point on, the discussion will be restricted to semisimple lie algebras, which are the ones of principal interest in physics. The cartan decomposition of a complex semisimple lie algebra shawn baland university of colorado, boulder november 29, 2007. Lie groups, lie algebras, and their representations university of. Pdf suppose g is a compact lie group and n is a closed normal subgroup of g acting freely on a smooth manifold x. Let g be a nite dimensional lie algebra over an algebraically closed eld f of characteristic zero and let h. However, applications of this theorem are still lacking. From an npov, the third lie theorem establishes the essential surjectivity of the functor lie lie from the category of local. The lie algebra version of gotos theorem 10 for compact semisimple lie groups is not as well known as it ought to be. For any zariskiconnected algebraic group, a cartan subgroup can be defined to be the centralizer of a maximal torus. Connected lie group an overview sciencedirect topics.
Then the group e l acts transitively on the set of borel subalgebrasof l. Cartans theorem states that the converse of this result is true at least for real lie groups. The equivalence between the category of simply connected real lie groups and finitedimensional real lie algebras is called usually in the literature of the second half of 20th century cartans or cartanlie theorem as it is proved by elie cartan whereas s. Introduction if a compact lie group g acts on a manifold m, the space mg of orbits of the action is usually a singular space.
Universal enveloping algebra and the casimir operator. On a theorem of henri cartan concerning the equivaraint. According to the previous lemma, one can nd a neighborhood uof ein gand a neighborhood v of 0 in g so that exp 1. Salamon eth zuric h 20 november 2019 contents 1 complex lie groups 2 2 first existence proof 5 3 second existence proof 8 4 hadamards theorem 16 5 cartans xed point theorem 18 6 cartan decomposition 20 7 matrix factorization 25 8 proof of the main theorems 31 1. Cartan s work on infinite dimensional lie al gebras, exterior differential calculus, differential ge ometry, and, above all, the representation theory of semisimple lie algebras was of supreme value. Commutators and cartan subalgebras in groups arxiv. H is subgroup of g which is also a closed subset of g. In particular, any borel subalgebra of a lie algebra lcontains a cartan subalgebra of l. We also show that maximal compact subgroups of the radical are contained in cartan subgroups, and for a connected solvable lie group, cartan subgroups are. The paper 1938a contains an interesting theorem on the approximation of lie groups by finite groups. The key to the proof of cartans theorem is to choose carefully a candidate subspace w. Let g be the complexi cation of g0, and choose a compact real form u0 of g. Equivariant cohomology and the cartan model eckhard meinrenken university of toronto 1. Suppose that g is the lie algebra of a lie group g.
We prove that if the maximal real rank is attained in the automorphism group of a geometry of. An embedding theorem for automorphism groups of cartan geometries. The original approach of cartan used riemannian geometry. Let l lieg be the lie algebra of a compact semisimple lie group g. Then the mapping x, k exp is a diffeomorphism of p x k onto g. Maximal tori by a torus we mean a compact connected abelian lie group, so a torus is a lie group that is isomorphic to tn rnzn. Proposition if d is a nilpotent derivation of g, then expd is an automorphism of g. In his third theorem, lie proved only the existence of of a local lie group, but not the global existence nor simply connected choice which were established a few decades later by elie cartan. We study the structure of cartan subgroups in a connected lie group and prove certain results which generalise the wustners structure theorem for cartan subgroups. Weyls complete reducibility theorem, levi and maltsev theorems. Pdf on a theorem of henri cartan concerning the equivaraint. Our embedding theorem leads to general upper bounds on the real rank or nilpotence degree of a lie subgroup of the automorphism group.