Representations
of Lie Groups and Lie Algebras |

**In General** > s.a. examples
of lie groups; lorentz and poincaré group; SU(2).

* __Approaches__: Infinitesimal
(Cartan-Weyl), and global; The latter gives simple canonical realizations of
the carrier space in terms of polynomials in complex variables.

* __Decomposition of tensor
products__: Formulas for multiplicites are
known for some cases, such as *n*-fold products of fundamental (*j* = 1/2) representations
of GL(*n*), U(*n*), SU(*n*); > s.a. Clebsh-Gordan
Coefficients.

* __Connected topological solvable__:
Every finite-D irr is 1D (Lie); __Examples__:
2D Poincaré group; Heisenberg group.

* __Semisimple connected__: Any representation is fully reducible.

* __Semisimple connected compact__:
There are no non-trivial finite-dimensional faithful unitary irr's.

* __Simple connected__: The only 1D irreducible representation is the trivial one.

* __Simple connected compact__:
There are no non-trivial finite-dimensional unitary irr's.

@ __Texts__: Varadarajan 84; Bröcker & tom Dieck 85; Fulton & Harris 91; Vilenkin & Klimyk 94 [and special functions].

@ __General references__: Chaturvedi et al RVMP(06) [Schwinger rep]; Kurnyavko & Shirokov a1710 [constructing infinitesimal invariants].

**Adjoint Representation**

$ __Def__: The mapping Ad: *G* → L(\(\cal G\), \(\cal G\)),
where \(\cal G\) is the Lie algebra of *G*, defined by Ad(*g*): \(\cal G\) →
\(\cal G\), corresponding
to (L_{g} R_{g}^{–1})'(*e*)
in the isomorphism between T_{e}*G* and \(\cal G\) given
by *α*_{e} ∈ T_{e}*G* (*α* =
left-invariant vector field generated by *α*_{e}).

* __Notation__: Often represented
by Ad(*g*) *γ* = *g* *γ* *g*^{–1},
literally correct if *G* is (a subgroup of) GL(*n*).

@ __References__: in Choquet-Bruhat et al 82.

**Specific Groups** > s.a. lie groups [generalizations].

* __SL(2, C)__:

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**Representations of Lie Algebras** > s.a. group representations /
lie algebras; simplex [polytope];
Special Functions.

$ __Adjoint representation__:
The differential of the adjoint representation of the Lie group *G* on \(\cal G\) at
the identity *e* (modulo the isomorphism
of T_{e}*G* onto \(\cal G\)),
ad: \(\cal G\) → L(\(\cal G\), \(\cal G\)),
defined by ad:= Ad'_{e}.

* __Matrix form__: If *f*_{ijk}
are the structure constants corresponding to the generators *T*_{i},
then (*T*_{i}^{adj})_{jk} = –i *f*_{ijk}.

@ __References__: Turbiner qa/97 [in
Fock space]; Humphreys BAMS(98)
[simple, modular representations]; de Azcárraga & Macfarlane
NPB(00)ht [fermionic];
Popovych et al JPA(03)mp [as
vector fields on a manifold]; Lau m.RT/04 [bosonic
and fermionic]; Henderson 12.

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