synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
A Lie group is a group with smooth structure. Lie groups form a category, LieGrp.
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.
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.
(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.
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 automorphism group of any Lie group is canonically itself a Lie group: the automorphism Lie group.
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 | 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 |
Glen Bredon, Section 0.5 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)
(in the broader context of topological groups)
Frank Adams, Lectures on Lie groups, University of Chicago Press, 1982 (ISBN:9780226005300, gbooks)
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]
Mark Haiman, lecture notes by Theo Johnson-Freyd, Lie groups, Berkeley 2009 (pdf)
Eckhard Meinrenken, Lie groups and Lie algebas, 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)
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]
In the generality of Lie semigroups:
In the generality of quantum 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)
On mapping spaces of continuous homomorphisms from topological groups to Lie groups:
for maps out of finitely generated discrete groups“:
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)
Last revised on September 7, 2023 at 12:27:14. See the history of this page for a list of all contributions to it.