nLab Lie group

Idea

A Lie group is a group with smooth structure. Lie groups form a category, LieGrp.

Definition

Definition

A Lie group is a smooth manifold whose underlying set of elements is equipped with the structure of a group such that the group multiplication and inverse-assigning functions are smooth functions.

In other words, a Lie group is a group object internal to the category SmthMfd of smooth manifolds.

Usually the smooth manifold is assumed to be defined over the real numbers and to be of finite dimension (f.d.), but extensions of the definition to some other ground fields or to -infinite-dimensional manifolds 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 called a compact Lie group (or connected, simply connected Lie group, etc) if its underlying topological space is compact (or connected, simply connected, etc).

Every connected finite dimensional 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.

Properties

Lie’s three theorems

Sophus Lie proved several theorems, known as Lie's three theorems, on the relationship between Lie algebras and Lie groups. Lie’s third theorem is about the equivalence of categories of finite-dimensional real Lie algebras and local Lie groups. Because Élie Cartan extended this to a global integrability theorem, Lie’s third theorem is also called the Cartan-Lie theorem.

Lie subgroups

Proposition

(Cartan's closed subgroup theorem)

If $H \subset G$ is a closed subgroup of a (finite dimensional) Lie group, then $H$ is a sub-Lie group, hence a smooth submanifold such that its group operations are smooth functions with respect to the the submanifold smooth structure.

Classification

Proposition

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

• classical Lie groups

and a finite snumber o

• exceptional Lie groups

Different Lie group structures on a group

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.

Example

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.

Different topologies on a Lie group

• Linus Kramer, The topology of a simple Lie group is essentially unique, (arXiv)

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.

Which topological groups admit Lie group structure?

• Hilbert's fifth problem

Relation to topological groups

• continuous homomorphisms of Lie groups are smooth

Homotopy groups

List of homotopy groups of the manifolds underlying the classical Lie groups are for instance in (Abanov 09).

Applications

In differential geometry

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 gauge theory

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

In higher category theory

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.

Examples

Basic examples

• 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 automorphism group of any Lie group is canonically itself a Lie group: the automorphism Lie group.

Classical Lie groups

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 $Sp(2n)$.

Exceptional Lie groups

The exceptional Lie groups incude

• G₂, F₄, E₆, E₇, E₈

Infinite-dimensional examples

• stable unitary group

• loop group

• diffeomorphism group

  • symplectomorphism group, quantomorphism group

Related concepts

• semisimple Lie group, simple Lie group, exceptional Lie group

• Lie monoid?, Lie groupoid, Lie category

• real form

• compact Lie group, maximal compact subgroup

• maximal torus

• rank of a Lie group

• conjugacy class, Cartan-Dirac structure on a Lie group

• Weyl group

• Inönü-Wigner group contraction

• infinite-dimensional Lie group

• orbit method

• Hamiltonian dynamics on Lie groups

• complex Lie group

• Poisson Lie group

• invariant differential form, Maurer-Cartan form

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		germ of a space		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

References

General

• Jean-Pierre Serre: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [doi:10.1007/978-3-540-70634-2]

• Glen Bredon, Section 0.5 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)

  (in the broader context of topological groups)

• Arthur A. Sagle, Ralph E. Walde: Introduction to Lie Groups and Lie Algebras, Pure and Applied Mathematics 51, Elsevier (1973) 215-227

• Nicolas Bourbaki, Lie groups and Lie algebras – Chapters 1-3, Springer (1975, 1989) [ISBN:9783540642428]

• Frank Adams, Lectures on Lie groups, University of Chicago Press, 1982 (ISBN:9780226005300, gbooks)

• M. M. Postnikov, Lectures on geometry: Semester V, Lie groups and algebras (1986) [ark:/13960/t4cp9jn4p]

• A. L. Onishchik (ed.) Lie Groups and Lie Algebras

  • I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,

  • II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups

  Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

• Tammo tom Dieck, Theodor Bröcker, Ch. I of: Representations of compact Lie groups, Springer 1985 (doi:10.1007/978-3-662-12918-0)

  (in the context of representation theory)

• José de Azcárraga, José M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge Monographs of Mathematical Physics, Cambridge University Press (1995) [doi:10.1017/CBO9780511599897]

• Howard Georgi, Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

  with an eye towards application to (the standard model of) particle physics

• Hans Duistermaat, Johan A. C. Kolk, Lie groups, Springer (2000) [doi:10.1007/978-3-642-56936-4]

• Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]

• Mark Haiman (notes by Theo Johnson-Freyd), Lie groups, lecture notes, Berkeley (2008) [pdf]

• Eckhard Meinrenken, Lie groups and Lie algebras, Lecture notes 2010 (pdf)

• Joachim Hilgert, Karl-Hermann Neeb, Structure and Geometry of Lie Groups, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 (doi:10.1007/978-0-387-84794-8)

• Daniel Bump, Lie groups, Graduate Texts in Mathematics 225, Springer (2013) [doi:10.1007/978-1-4614-8024-2, pdf]

• Brian C. Hall, Lie Groups, Lie Algebras, and Representations, Springer 2015 (doi:10.1007/978-3-319-13467-3)

• Jean Gallier, Jocelyn Quaintance, Differential Geometry and Lie Groups – A computational perspective, Geometry and Computing 12, Springer (2020) [doi:10.1007/978-3-030-46040-2, webpage]

• Jean Gallier, Jocelyn Quaintance, Differential Geometry and Lie Groups – A second course, Geometry and Computing 13, Springer (2020) [doi:10.1007/978-3-030-46047-1, webpage]

• Pavel Etingof, Lie groups and Lie algebras [arXiv:2201.09397]

In the generality of Lie semigroups:

• Joachim Hilgert, Karl-Hermann Neeb, Lie Semigroups and their Applications. Lecture Notes in Mathematics 1552, Springer 1993 (doi:10.1007/BFb0084640)

In the generality of quantum groups:

• Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin, Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (2020-2024) [pdf]

Homotopy groups

• Alexander Abanov, Homotopy groups of Lie groups (2009) [pdf]

On infinite-dimensional Lie groups

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, Infinite dimensional Lie Theory from the point of view of Functional Analysis (pdf)

• Karl-Hermann Neeb, Monastir summer school: Infinite-dimensional Lie groups (pdf)

Spaces of homomorphisms

On mapping spaces of continuous homomorphisms from topological groups to Lie groups:

for maps out of finitely generated discrete groups“:

• Frederick R. Cohen, Mentor Stafa, A survey on spaces of homomorphisms to Lie groups, In: Callegaro F., Cohen F., De Concini C., Feichtner E., Gaiffi G., Salvetti M. (eds.) Configuration Spaces, INdAM Series, vol 14, Springer 2016 (arXiv:1412.5668, doi:10.1007/978-3-319-31580-5_15)

for maps out of compact Lie groups and the fact that nearby homomorphisms from compact Lie groups are conjugate:

• Pierre Conner, Edwin Floyd, Ch. III, Lem. 38.1 in: Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete 33, Springer 1964 (doi:10.1007/978-3-662-41633-4)

• Charles Rezk, Nearby homomorphisms from compact Lie groups are conjugate (MO:q/123624)

• Charles Rezk, Rem. 2.2.1 in: Global Homotopy Theory and Cohesion, 2014 (pdf, pdf)