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 Cinfty manifold,
and the group operations are smooth,
a 
G the
maps G → G by b 
ba and b b–1 are
Cinfty .
* Applications: They often appear as diffeomorphisms on a manifold X (> see smooth realizations).
Specific Concepts
> see Center;
coordinates; Flag
Manifold; differential form [canonical]; Homogeneous
Space; Left Translation;
Right Action.
Additional Structures > s.a. measure [Haar].
* 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: 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) = R ×
S3.
@ Integration: Collins & Sniady CMP(06)mp/04 [on
U(n), O(n)
and Sp(n)]; > s.a. examples of lie groups.
@ Related topics: Szarek m.FA/97 [Finsler
geometry]; Ghanam et al JMP(07) [metrics].
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,R) = O(n) × C, where C =
{positive-definite symmetric matrices}; GL(n,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 cy, with no fuss about differentiability or global
topology.
@ References: Bochner AM(46); Dieudonné 73; Hazewinkel 78.
Inhomogeneous Groups > s.a. poincaré [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 ILambda G,
depending
on
a parameter
R.
@ References: Romano GRG(93)gq and
refs.
References > s.a. BRST; Casimir
Operator; quantum mechanics; representations;
Special Functions.
@
Texts: Eisenhart 33; Weyl 46; Chevalley 46; Serre 64; Warner 71; Bourbaki
73; Helgason 78; Onishchik & Vinberg 90; Hall mp/00-ln;
Knapp 02 [IV]; Rossmann 02; Hall 06.
@ For physicists: Lipkin 65; Hermann 66; Azcárraga & Izquierdo
95; Kolev mp/04-in
[use in mechanics]; Fecko 06.
@ Infinite-dimensional: Milnor in(84).
@ Classical groups: Kleidman & Liebeck 90.
@ Related topics: Schmidt JMP(87)
[topology on space of G's]; 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].
@ Lie 2-groups: Baez ht/02 [and
higher Yang-Mills theory].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
21 jun 2008