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* ∈ T*G* 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) = \(\mathbb R\) × sS^{3}.

@ __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 *g*_{ab}:=
*C*^{c}_{da}
*C*^{d}_{cb},
where *C*^{a}_{bc}
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 *I*_{Λ} *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 *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];
> s.a gauge theories.

@ __Lie 2-groups__: Baez ht/02 [and higher Yang-Mills theory].

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