G is representation of a lie group, with elements u that are unitary matrices of size. Lectures on lie groups and representations of locally compact. Finally, we show that if g is a lie group endowed with a bi invariant. Razavi where z is a nonzero vector in the lie algebra g of g. The parametrization of the group introduces an explicit construction of the haar measure z g dufu z dnx p detgfx 3. Concentration of measure and the compact classical matrix. Homogeneous geodesics of left invariant randers metrics. Having an invariant measure on gh means that the density bundle is equivariantly trivial.
An invariant random subgroup irs of gis a random element of sub gwhose law is a conjugation invariant borel probability measure. Ratner proved in a series of papers 32, 31, 26, 27, 33 very strong results on invariant measures and orbit closures for certain subgroups hof a lie group g where h acts on the right of x. Similarly, the measure of hyperbolic angle is invariant under squeeze mapping. The corresponding measure is called haar measure or left haar measure. State estimation for invariant systems on lie groups with. More generally, consider ga semisimple lie group with. If a connected lie group g has a left invariant metric with all sectional curvatures k invariant metrics and killing forms are then discussed, e. Ratner proved in a series of papers 34, 33, 28, 29, 35 very strong results on invariant measures and orbit closures for certain subgroups h of a lie group g where h acts on the right of x. Kajzer in 3 where he proved that a lie group g with a leftinvariant metric has at least one homogeneous geodesic through the identity. Namely, we establish the formulas giving di erent curvatures at the level of the associated lie algebras. Every locally compact group has a haar measure that is invariant under the group action. Let f 1be the group with in nite, countably many generators. Finally, an invariant measure for a family of measurable transformations, such as a semi group, a group, a flow, etc.
Recall that a lie group is called uni modular if its left invariant haar measure is also right invariant section 6. We first give a necessary and sufficient condition for a left. 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. Left invariant volume form exists on any lie group g, and is unique up to a multiplicative constant. Invariant markov processes under lie group actions will be of interest to researchers in stochastic analysis and probability theory, and will also appeal to experts in lie groups, differential geometry, and related topics interested in applications of their own subjects.
Invariant functions on lie groups and hamiltonian flows. The theorem is true in much more generality in particular, any compact lie group has a haar probability measure, but we wont worry about that, or the proof of the theorem. We denote again 2010 mathematics subject classi cation. Pdf in this paper we study the geometry of lie groups with biinvariant randers metric. Invariant markov processes under lie group actions ming. More precisely, decomposing endg into the direct sum of the subspaces consisting of all endomorphisms of g which are selfadjoint or, respec. Then there exists a unique left invariant pseudoriemannian structure q on g such that q b. Thus 1 is an eigenvalue for p and therefore also for p writing a p.
Invariant measures and the set of exceptions to littlewood. Area measure in the euclidean plane is invariant under 2. The additive group rn has left and right invariant haar measure, namely the lebesgue measure d dx 1dx 2 dx n. Thompson, killings equations for invariant metrics on lie groups, journal of geometry and mechanics, 3 2011, 323. When g is a matrix lie group, this product is simply a matrix multiplication. Let b be a nondegenerate symmetric bilinear form on g x g. As a corollary, it follows that any direct products of an abelian lie group of dimension 1 and a nonabelian lie group also have nonunique metrics. Invariant functions on lie groups and hamiltonian flows ofsurface group representations william m. Let gbe a product of a totally disconnected group and a lie group.
Let gbe a locally compact, second countable group and let sub g be the space of closed subgroups of g, considered with the chabauty topology 10. Invariant distributions, statement of existence and uniqueness up to constant multiples. Lecture notes introduction to lie groups mathematics. The haar measure of a lie group a simple construction. Homogeneous geodesic in a lie group were studied by v. State estimation for invariant systems on lie groups with delayed output measurements. The notion of an invariant measure plays an important role in the theory of dynamical systems and ergodic theory. 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.
Therefore, if we understand the latter measures we understand all of them. For example the uncertainty principle related to lie groups presented in 1 does not include any statements about an invariant measure, but the measure plays an important role in proving the theorem. Concentration of measure and the compact classical matrix groups. The general linear group over the real numbers, denoted by gln. If g is a lie group, then an elementary calculation shows that.
Ratner showed that all hinvariant and ergodic probability measures are of the form ml. This book is essentially a written version of those lectures. We study also the particular case of bi invariant riemannian metrics. On a lie group we can construct an invariant measure from the lebesgue measure on the lie algebra. Curvatures of left invariant metrics on lie groups john milnor. Assume l has finite center, and that the real rank of every simple factor of l is at least. We study also the particular case of biinvariant riemannian metrics. Here are some of the basic facts about invariant measures on homogeneous spaces for locally. Invariant measures and arithmetic quantum unique ergodicity. Then there is a countable ordinal 1 3 such that the set eqinv f 1 is 0 1complete.
If a connected lie group g has a left invariant metric with all sectional curvatures k pdf in this paper we study the geometry of lie groups with biinvariant randers metric. Curvatures of left invariant metrics on lie groups john. Other work in this area is the consideration of multiple output maps formulated by actions of the lie group on homogeneous output spaces and also adaptive estimation of an unknown input bias. The explanation of the general concept of the group invariant haar measure can be found for instance in the book theory of group representations and applications by barut and raczka. Let dx be the left haar measure on g which is unique up to.
If is a discrete group with the discrete topology, then. In this paper, we study the geometry of lie groups with bi invariant finsler metrics. Then there is a unique up to constant multiples right g invariant continuous linear functional uon the space of test functions c1 c g on g, namely an integral uf z g fxdx 2. Curvature of left invariant riemannian metrics on lie groups. 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. If g is a semigroup and p a metric on g, p will be called left invariant if pgx, gy px, y whenever g, x, y cg, right invariant if always pxg, yg px, y, and invariant if it is both right and left invariant. Cocycle superrigidity for ergodic actions of nonsemisimple. The lectures began with general measure theory and went on to haar measure and some of its generalizations. The haar measure of a lie group a simple construction l.
This conjecture is a special case of much more general conjectures in this. Invariant control systems on lie groups rory biggs claudiu c. On a lie group we can construct an invariant measure from the. Also, in general one has to worry about invariance under lefttranslation or right. Left invariant connections ron g are the same as bilinear. Then i if h is compact, gh has an invariant measure. Introduction to lie groups, lie algebras and their representations. Quasiinvariant measures for continuous group actions. The volume element for the haar measure is constructed by means of the metric, which enters in the invariant measure. Tamaru, the space of left invariant metrics on a lie group up to isometry and scaling, manuscripta math. Invariant measures are important tools in many areas of mathematics.
Goldman department ofmathematics, massachusetts institute oftechnology, cambridge, ma029, usa in7itwas shownthatifn is thefundamental groupofa closed orientedsurface s and gis lie groupsatisfying very general conditions, then the space homn,gg. Then, we prove that every compact connected lie group is a symmetric finsler space with respect to the bi invariant absolute homogeneous finsler metric. Primary 03c, secondary 03c15, 05d10, 37b05, 37a15, 54h20. On the existence of biinvariant finsler metrics on lie.
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