Topological group
Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a construct that can be defined on a very wide class of topological groups.
To show that a topology is compatible with the group operations, it suffices to check that the map is continuous.
Explicitly, this means that for any x, y ∈ G and any neighborhood W in G of xy−1, there exist neighborhoods U of x and V of y in G such that U ⋅ (V−1) ⊆ W. This definition used notation for multiplicative groups; the equivalent for additive groups would be that the following two operations are continuous: Although not part of this definition, many authors[3] require that the topology on G be Hausdorff.
Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.
p of p-adic integers, for a prime number p, meaning the inverse limit of the finite groups
p is well behaved in that it is compact (in fact, homeomorphic to the Cantor set), but it differs from (real) Lie groups in that it is totally disconnected.
For example, the group of invertible bounded operators on a Hilbert space arises this way.
[4] Thus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.
If G is a locally compact commutative group, then for any neighborhood N in G of the identity element, there exists a symmetric relatively compact neighborhood M of the identity element such that cl M ⊆ N (where cl M is symmetric as well).
If U is an open subset of a commutative topological group G and U contains a compact set K, then there exists a neighborhood N of the identity element such that KN ⊆ U.
[8] An alternative approach was made by Uffe Haagerup and Agata Przybyszewska in 2006,[9] the idea of the which is as follows: One relies on the construction of a left-invariant metric,
Closing the open ball, U, of radius 1 under multiplication yields a clopen subgroup, H, of G, on which the metric
For example, for a positive integer n, the sphere Sn is a homogeneous space for the rotation group SO(n+1) in
[4] Every discrete subgroup of a Hausdorff commutative topological group is closed.
[4] The isomorphism theorems from ordinary group theory are not always true in the topological setting.
For example, a native version of the first isomorphism theorem is false for topological groups: if
is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not necessarily be a homeomorphism.
[12] The third isomorphism theorem, however, is true more or less verbatim for topological groups, as one may easily check.
As shown by Andrew Gleason, Deane Montgomery, and Leo Zippin, the answer to this problem is yes.
[16] For example, the theory of Fourier series describes the decomposition of the unitary representation of the circle group S1 on the complex Hilbert space L2(S1).
The irreducible representations of S1 are all 1-dimensional, of the form z ↦ zn for integers n (where S1 is viewed as a subgroup of the multiplicative group
[17]) A basic example is the Fourier transform, which decomposes the action of the additive group
For a locally compact abelian group G, every irreducible unitary representation has dimension 1.
Pontryagin duality states that for a locally compact abelian group G, the dual of
[20] In particular, for a connected Lie group G, the rational cohomology ring of G is an exterior algebra on generators of odd degree.
Moreover, a connected Lie group G has a maximal compact subgroup K, which is unique up to conjugation, and the inclusion of K into G is a homotopy equivalence.
Since the hyperbolic plane is contractible, the inclusion of the circle group into SL(2,
Finally, compact connected Lie groups have been classified by Wilhelm Killing, Élie Cartan, and Hermann Weyl.
As a result, there is an essentially complete description of the possible homotopy types of Lie groups.
and similarly their difference is defined to be the image of the product net under the subtraction map:
