Representations of Lie Groups and Lie Algebras| 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 (Lg
Rg−1)'(e)
in the isomorphism between TeG and \(\cal G\)
given by αe ∈
TeG (α = left-invariant
vector field generated by αe).
* 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):
@ SL(n, R):
Friedman & Sorkin JMP(80) [SL(4, \(\mathbb R\))];
Sijacki JMP(90) [SL(3, \(\mathbb R\)) ladder representations];
Basu ht/01 [principal series].
@ SU(n): García & Perelomov JPA(02)mp [characters];
Shurtleff a0908 [SU(3), formulas for matrixes].
@ SO(n): Bargmann RMP(62);
Lorente & Kramer gq/04-conf,
a0804 [SO(4) and quantum gravity].
@ Other:
Bars & Teng JMP(90) [SU(2,1), unitary irr's];
Boya RPMP(93) [simple groups];
Barnea JMP(99) [O(n), recursive];
Ibort et al a1610
[compact Lie groups, numerical algorithm to decompose unitary representations].
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
TeG onto \(\cal G\)),
ad: \(\cal G\) → L(\(\cal G\), \(\cal G\)),
defined by ad:= Ad'e.
* Matrix form: If fijk
are the structure constants corresponding to the generators Ti,
then (Tiadj)\(_{jk}\)
= −i fijk.
@ 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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