Usually the manifold is assumed to be over the real numbers or the complex numbers and of finite dimension (f.d.), but extensions to some other ground fields and infinite-dimensional setting are also relevant, sometimes under other names (such as Fréchet Lie group when the underlying manifold is an infinite-dimensional Fréchet manifold).
There is an infinitesimal version of a Lie group, a so-called local Lie group, where the multiplication and the inverse are only partially defined, namely if the arguments of these operations are in a sufficiently small neighborhood of identity. There is a natural equivalence of local Lie groups by means of agreeing (topologically and algebraically) on a smaller neighborhood of the identity. The category of local Lie groups is equivalent to the category of connected and simply connected Lie groups.
Sophus Lie has proved several theorems – Lie's three theorems – on the relationship between Lie algebras and Lie groups. What is called Lie's third theorem is about the equivalence of categories of f.d. real Lie algebras and local Lie groups. Élie Cartan has extended this to a global integrability theorem called the Cartan-Lie theorem, nowadays after Serre also called Lie’s third theorem.
Every connected finite-dimensional real Lie group is homeomorphic to a product of a compact Lie group and a Euclidean space. Every abelian connected compact f.d. real Lie group is a torus (a product of circles ).
The simple Lie groups have a classification into infinite series of
and a finite snumber o
Abstract: We study locally compact group topologies on simple Lie groups. We show that the Lie group topology on such a group is very rigid: every ‘abstract’ isomorphism between and a locally compact and -compact group is automatically a homeomorphism, provided that is absolutely simple. If is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.
In the physics of gauge fields – gauge theory – Lie groups appear as local gauge groups parameterizing gauge transformations: notably the Yang-Mills field is modeled by a -principal bundle with connection for some Lie group . For models that describe experimental observations the group in question is a quotient of a product of special unitary groups and the circle group. For details see standard model of particle physics
The classical Lie groups include
the symplectic group .
The exceptional Lie groups incude
Examples of sequences of infinitesimal and local structures
|geometry||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||p-adic integers||integers|
|Lie theory||Lie algebra||formal group||local Lie group||Lie group|
|symplectic geometry||Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|
References on infinite-dimensional Lie groups
Rudolf Schmid, Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics Advances in Mathematical Physics Volume 2010, (pdf)
Josef Teichmann, Innite dimensional Lie Theory from the point of view of Functional Analysis (pdf)