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,

aG   the maps   GG 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.
@ Physics applications: Kolev mp/04-proc [mechanics]; Öttinger JNFM(10)-a1002 [non-equilibrium thermodynamics]; Dahm a1102-conf [hadron physics and spacetime]; Paliathanasis & Tsamparlis IJGMP-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 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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