 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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