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.

@ __Physics applications__: Kolev mp/04-proc
[mechanics]; Öttinger JNFM(10)-a1002 [non-equilibrium thermodynamics]; Dahm a1102-conf [hadron physics and spacetime]; Paliathanasis & Tsamparlis IJGMP(14)-a1312 [Lie point symmetries of the Schrödinger and Klein-Gordon equations];
> 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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