A Lie group is a group with smooth structure.
A Lie group is a smooth manifold whose underlying set of points is equipped with a structure of a group and the multiplication and inverse maps for the group are smooth maps.
In other words, it is a group object internal to Diff.
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).
A real Lie group is a compact Lie group (or connected, simply connected Lie group, etc) if its underlying space is compact (or connected, simply connected, etc).
Every connected finite dimensiona real Lie group is homeomorphic to a product of a compact Lie group (its maximal compact subgroup) and a Euclidean space.
Every abelian connected compact finite dimensional real Lie group is a torus (a product of circles $T^n = S^1\times S^1 \times \ldots \times S^1$).
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.
The first order infinitesimal approximation to a Lie group is its Lie algebra.
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 $T^n = S^1\times S^1 \times \ldots \times S^1$).
The simple Lie groups have a classification into infinite series of
and a finite snumber o
For $G$ a bare group (without smooth structure) there may be more than one way to equip it with the structure of a Lie group.
As bare abelian groups, the Cartesian spaces $\mathbb{R}^n$ are, for all $n$, vector spaces over the rational numbers $\mathbb{Q}$ whose dimension is the cardinality of the continuum, $2^{\aleph_0}$.
Therefore these are all isomorphic as bare group. But equipped with their canonical Lie group structure (as in the Examples) they are of course not isomorphic.
Abstract: We study locally compact group topologies on simple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every ‘abstract’ isomorphism between $S$ and a locally compact and $\sigma$-compact group $\Gamma$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.
List of homotopy groups of the manifolds underlying the classical Lie groups are for instance in (Abanov 09).
A central concept of differential geometry is that of a $G$-principal bundle $P \to X$ over a smooth manifold $X$ for $G$ a Lie group.
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 $G$-principal bundle with connection for some Lie group $G$. For models that describe experimental observations the group $G$ 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 notion of group generalizes in higher category theory to that of 2-group, … ∞-group.
Accordingly, so does the notion of Lie group generalize to Lie 2-group, … ∞-Lie group. For details see ∞-Lie groupoid.
The real line $\mathbb{R}$ with its standard smooth structure and the group operation being addition is a Lie group. So is every Cartesian space $\mathbb{R}^n$ with the componentwise addition of real numbers.
The quotient of $\mathbb{R}$ by the subgroup of integers $\mathbb{Z} \hookrightarrow \mathbb{R}$ is the circle group $S^1 = \mathbb{R}/\mathbb{Z}$. The quotient $\mathbb{R}^n/\mathbb{Z}^n$ is the $n$-dimensional torus.
The classical Lie groups include
the general linear group $GL(n)$
the orthogonal group $O(n)$ and special orthogonal group $SO(n)$;
the unitary group $U(n)$ and special unitary group $SU(n)$;
the symplectic group $Sp(2n)$.
The exceptional Lie groups incude
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
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 | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ 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 |
Mark Haiman, lecture notes by Theo Johnson-Freyd, Lie groups, Berkeley 2009 (pdf)
Eckhard Meinrenken, Lie groups and Lie algebas, Lecture notes 2010 (pdf)
References on infinite-dimensional Lie groups
Andreas Kriegl, Peter Michor, Regular infinite dimensional Lie groups Journal of Lie Theory Volume 7 (1997) 61-99 (pdf)
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)
Karl-Hermann Neeb, Monastir summer school: Infinite-dimensional Lie groups (pdf)