Lie Groups| Lie Groups |
In General > s.a. examples of lie groups.
* Idea: A group whose elements
depend continuously on a set of parameters, compatibly with the group structure.
$ Def: A Lie group is a group
G which is also a C∞ manifold,
and the group operations are smooth,
∀a ∈ G the maps G → G by b \(\mapsto\) ba and b \(\mapsto\) b−1 are C∞ .
* Applications:
They often appear as diffeomorphisms on a manifold X
(> see smooth realizations).
> Online resources:
see Wikipedia page.
Specific Concepts
> see Center;
coordinates; Flag Manifold;
differential form [canonical]; Homogeneous
Space; Left Translation; Right Action.
Additional Structures and Constructions
> s.a. measure [Haar]; tangent structures.
* Invariant vector fields:
We can get a vector field X ∈ TG which is
left-invariant (X(hg) = X(h)) and one
which is right-invariant (X(gh) = X(h))
from every generator of the group.
* One-parameter subgroup:
A differentiable curve g: \(\mathbb R\) → G,
such that g(t) g(s)
= g(t+s), and g(0) = e.
* Metric: Given a left-invariant
measure, a left-invariant metric on a compact Lie group can be obtained by
averaging any given metric over the group; s.a. Semisimple groups below.
* Lorentzian metric: There is one
essentially only on SO(2,1) = \(\mathbb R\) × sS3.
@ Integration: Collins & Sniady CMP(06)mp/04
[on U(n), O(n) and Sp(n)];
> s.a. examples of lie groups.
@ Metrics: Ghanam et al JMP(07);
Pope a1001 [homogeneous Einstein metrics];
Hervik a1002-conf [negatively curved, left-invariant].
@ Related topics:
Szarek m.FA/97 [Finsler geometry];
> s.a. Central Extension.
Compact Lie groups
* Result: They
are all isomorphic to some subgroup of some O(m).
@ References: Fegan 91;
Boya RPMP(91) [geometry].
Connected Lie groups
* Result: They are all of
the form G = H × D, where H is a
maximal compact subgroup, and D is a topologically Euclidean space;
In particular: GL(n, \(\mathbb R\)) = O(n) × C,
where C = {positive-definite symmetric matrices}; GL(n,
\(\mathbb C\)) = U(n) × C, where C
= {positive-definite hermitian matrices}.
Semisimple Lie groups
* Metric: A natural
one is gab:=
Ccda
Cdcb,
where Cabc
are the structure constants of the group; It is left- and right-invariant. (Is this the Cartan-Killing metric?)
@ References:
Nevo & Zimmer AM(02) [actions].
Formal Groups
* Idea: Lie groups
treated in the style of the XVIII century, with no fuss about
differentiability or global topology.
@ References: Bochner AM(46);
Dieudonné 73;
Hazewinkel 78.
Inhomogeneous Groups
> s.a. poincaré group [ISO(3,1)].
$ Inhomogeneous extensions:
Given a Lie group G, its inhomogeneous extension IG,
as a manifold, is T*G, and as a group the semidirect product of
G with an Abelian group of the same dimension as G.
* Inhomogeneous Lorentz:
ISO(p, q) = SO(p, q)
⊗ T(p+q).
* Generalization: Can generalize
to IΛ G, depending
on a parameter Λ ∈ \(\mathbb R\).
@ References:
Romano GRG(93)gq and refs.
References
> s.a. BRST transformations; Casimir
Operator; quantum mechanics; representations;
Special Functions.
@ Texts: Eisenhart 33;
Weyl 46;
Chevalley 46;
Serre 64;
Warner 71;
Bourbaki 75;
Helgason 78;
Onishchik & Vinberg 90;
Hsiang 00;
Hall mp/00-ln;
Knapp 02 [IV];
Rossmann 02;
Duistermaat & Kolk 04;
Hall 06;
Abbaspour & Moskowitz 07;
Procesi 07 [through invariants and representations;
r BAMS(08)];
Ivancevic & Ivancevic a1104-ln;
Steeb et al 12 [problems and solutions].
@ For physicists: Lipkin 65;
Hermann 66;
Azcárraga & Izquierdo 95;
Fecko 06; Gilmore 08;
Huang a2012 [tutorial].
@ Physics applications: Kolev mp/04-proc [mechanics];
Öttinger JNFM(10)-a1002 [non-equilibrium thermodynamics];
Dahm PAN(12)-a1102-conf [hadron physics and spacetime];
Paliathanasis & Tsamparlis IJGMP(14)-a1312
[Lie point symmetries of the Schrödinger and Klein-Gordon equations];
Celeghini et al a1907 [and special functions and rigged Hilbert spaces];
> s.a. conservation laws; cosmological models.
@ Infinite-dimensional: Milnor in(84).
@ Classical groups: Kleidman & Liebeck 90.
@ Related topics:
Schmidt JMP(87) [topology on the space of Gs];
Sabinin 04 [mirror geometry];
> s.a. lagrangian systems.
Generalizations > s.a. quantum group.
* Lie groupoid: It canonically defines
both a C*-algebra C*(G) and a Poisson manifold A*(G).
* Lie 2-group: A category
C where the set of objects and the set of morphisms are Lie groups,
and the source, target, identity and composition maps are homomorphisms.
@ Super Lie groups: Carmeli et al CMP(06)ht/05 [unitary representationss and applications].
@ Homotopy Lie groups:
Møller BAMS(95).
@ Lie groupoids: Landsman & Razaman mp/00 [associated Poisson algebras];
Landsman JGP(06)mp/05 [in physics, rev];
> s.a gauge theories.
@ Lie 2-groups: Baez ht/02 [and higher Yang-Mills theory].
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