Orthogonal and Unitary Lie Groups and Lie Algebras  

Orthogonal Groups > s.a. examples of lie groups [relationships]; fundamental groups; grand unified theories [SO(10)].
* O(n): The group of invertible linear operators on the n-dimensional real vector space which preserve the Euclidean form; In an orthonormal basis for the vector space, they are given by orthogonal matrices M, satisfying M–1 = MT; The dimension of the group is n (n–1)/2.
* SO(n): The subgroup of O(n) of SO(n) of orthogonal matrices with unit determinant; Its dimension is n (n–1)/2 (it is the component of SO(n) connected to the identity).
* SO(3): ≅ \(\mathbb R\)P3, locally the same as SU(2), simple; > s.a. rotations.
* SO(4): Isomorphic to (SU(2) × SU(2))/\(\mathbb Z\)2 [@ see Thurston 97 for details]; Topologically, SO(4) = (S3 × S3)/{(1,1), (–1,–1)}, where {(1,1), (–1,–1)} = ker(h), with h: S3 × S3 → SO(4) is the surjective homorphism given by h(p,q)(x):= p–1xq, in which x ∈ \(\mathbb R\)4, we have identified p, q ∈ S3 with unit quaternions, and multiplication is quaternion multiplication; Also homeomorphic to SO(3) × SU(2) = P3 × S3, but this cannot be made into a Lie group equivalence.
@ General references: Zhang a1509 [volumes]; Diaconis & Forrester a1512 [measure, history].
@ O(n): Gorin JMP(02)mp/01, Braun JPA(06)mp [integrals, > s.a. lie groups].
@ SO(3): Mebius math/07 [derivation of the Euler-Rodrigues formula]; Mukunda et al a0904 [Hamilton's theory of turns]; > s.a. SU(2).
@ SO(4): Mebius www(01), math/05 [quaternion representation theorem]; > s.a. Wikipedia page.
@ SO(n): Alisauskas JPA(02)mp [3j symbols], JPA(02) [6j symbols]; Jiang & Soudry AM(03) [local converse theorem for SO(2n+1)].
> Online resources: see Wikipedia page.

Pseudo-Orthogonal Groups > s.a. lorentz group [SO(3,1)]; de sitter group [SO(4,1)] / fundamental groups; Racah Coefficients.
* SO(p, q): Non-compact; SO(2,1) = SL(2,\(\mathbb R\))/\(\mathbb Z\)2; SO(3,1) is simple; SO(2,2) = SL(2,\(\mathbb R\)) × SL(2,\(\mathbb R\)).
* SO(3,1) Lie algebra: The generators are the rotations Si and boosts Ki ,

\[\matrix{
S_1 = \left(\matrix{0&0&0&0\cr 0&0&0&0\cr 0&0&0&-1\cr 0&0&1&0}\right)
&S_2 = \left(\matrix{0&0&0&0\cr 0&0&0&1\cr 0&0&0&0\cr 0&-1&0&0}\right)
&S_3 = \left(\matrix{0&0&0&0\cr 0&0&-1&0\cr 0&1&0&0\cr 0&0&0&0}\right)\cr
K_1 = \left(\matrix{0&1&0&0\cr 1&0&0&0\cr 0&0&0&0\cr 0&0&0&0}\right)
&K_2 = \left(\matrix{0&0&1&0\cr 0&0&0&0\cr 1&0&0&0\cr 0&0&0&0}\right)
&K_3 = \left(\matrix{0&0&0&1\cr 0&0&0&0\cr 0&0&0&0\cr 1&0&0&0}\right) \;,
} \]

satisfying [Si, Sj] = εijk Sk , [Si, Kj] = εijk Kk , and [Ki, Kj] = –εijk Sk .
* SO(2,1) Lie algebra: The generators are T0, T1 and T2, with commutators [Ti, Tj] = fijk Tk = εijk gkl Tl, where ε012 = 1, and gij = diag(–1,1,1) = \(1\over2\)fikl fjlk.
@ References: Alhaidari PRA(02)mp/01 [SO(2,1), graded extension and physics]; Jafari & Shariati PRD(11)-a1109 [projective actions and doubly-special relativity].

Unitary Groups > s.a. holonomy [U(1)]; lie algebra; representations; standard model; SU(2); yang-mills theories.
* U(n): The dimension is n2; Simple.
* U(2): tr(AB) + tr(AB–1) = (tr A) (tr B).
* SU(n): The dimension is n2–1; The rank of SU(4) is 3.
* U(∞) and SU(∞): Inductive limits of U(n) and SU(n), respectively.
@ General references: Spengler et al JMP(12)-a1103 [composite parametrization and Haar measure]; Zhang a1509 [volumes].
@ SU(3): Gsponer mp/02/JMP [quaternionic parametriz]; Kerner a0901 [from \(\mathbb Z\)3- graded cubic algebra]; Shurtleff a0908 [formulas for matrices]; Shurtleff a1001 [and the 8D Poincaré group]; Grimus & Ludl JPA(10)-a1006 [subgroups]; Ludl JPA(11)-a1101 [classification of finite subgroups].
@ SU(4): Tilma et al JPA(02)mp [Euler angle parametriz]; Gsponer mp/02/JMP [quaternionic parametriz].
@ SU(n): Rudolph & Schmidt mp/01 [orbits on compact M]; Tilma & Sudarshan JPA(02)mp [Haar measure, Euler angles]; Bertini et al JMP(06)mp/05 [Euler angles]; Akhtarshenas a1003 [invariant vector fields and one-forms]; Shurtleff a1009 [and rotations, boosts, and translations in N 2-dimensional spacetime]; Mujtaba JGP(12) [homogeneous Einstein metrics].
@ U(n): Tilma & Sudarshan JGP(04)mp/02 [Euler angles]; Aubert & Lam JMP(03)mp, JMP(04)mp [integration]; Spengler et al JPA(10)-a1004 [parametrization].
@ U(∞) and SU(∞): in Mavromatos & Winstanley CQG(00)ht/99; Borodin & Olshanski AM(05)m.RT/01 [harmonic analysis]; Swain ht/04, ht/04, ht/04 [SU(∞) is not isomorphic to SDiff(2M)].
@ Related topics: Croxson PLA(06)qp/04 [SU(2), SU(2,1) and t-dependent Hamiltonians].

Pseudo-Unitary Groups > s.a. hamiltonian systems [SU(1,1)]; lie algebras.
* U(p, q):
* SU(p, q): Non-compact.
* SU(1, 1): 3D; Casimir invariant C2 = K32K12K22, with eigenvalues \(\hbar\)2k(k–1) (discrete) and \(\hbar\)2(–λ2–1/4) (continuous); Can be parametrized by α, β ∈ \(\mathbb C\) with |α|2 + |β|2 = 1, for example as

\[U = \left(\matrix{\alpha&\beta\cr\beta^*&\alpha^*}\right),\quad \alpha = \cosh(\tau/2)\,{\rm e}^{-{\rm i}\nu_1}\;,\quad \beta = \sinh(\tau/2)\,{\rm e}^{-{\rm i}\nu_2}\;, \quad \tau > 0\;, \quad \nu_i\in[0,2\pi]\;.\]


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