Representations
of Lie Groups and Lie Algebras| Representations
of Lie Groups and Lie Algebras |
Of a Lie Group > s.a. 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:
No non-trivial finite-dimensional faithful uirr's.
* Simple connected: The only 1D irr is the trivial one.
* Simple connected compact: No non-trivial finite-D uirr's.
Adjoint Representation
$ Def: The mapping Ad: G → L(
, ),
where is
the Lie algebra of G, defined by Ad(g): →
, corresponding
to (Lg Rg–1)'(e)
in the isomorphism between TeG and given
by 
e 
TeG ( =
left-invariant vector field generated by e).
* Notation: Often represented
by Ad(g) 
= g g–1,
literally correct if G is (a subgroup of) GL(n).
@ References: in Choquet-Bruhat et al 82.
References > s.a. lie groups [generalizations].
@ General references: Varadarajan 84; Bröcker & tom Dieck 85; Fulton & Harris
91; Vilenkin & Klimyk 91–92 [and special functions].
@ SL(n,R): Friedman & Sorkin JMP(80)
[SL(4,R)];
Sijacki JMP(90)
[SL(3,R) ladder representations]; Basu ht/01 [principal
series of representations].
@ SU(n): García & Perelomov mp/02 [characters].
@ SO(n): Bargmann RMP(62);
Lorente & Kramer gq/04-in,
a0804 [SO(4) and quantum gravity].
@ Other: Bars & Teng JMP(90)
[SU(2,1), uirr's]; Boya RPMP(93) [simple groups]; Barnea JMP(99)
[O(n),
recursive]; Chaturvedi et al RVMP(06)
[Schwinger rep].
Of a Lie Algebra > s.a. [representations];
lie algebras; simplex [polytope];
Special Functions.
$ Adjoint representation:
The differential of the adjoint representation of the Lie group G on at
the identity e (modulo the isomorphism
of TeG onto ),
ad: →
L(, ),
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 reps]; 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].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
27 may 2008