Orthogonal and Unitary Lie Groups and Lie Algebras| 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 parametrizations];
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];
Roelfs a2102 [novel invariant decomposition].
@ SU(4): Tilma et al JPA(02)mp [Euler angle parametriz];
Gsponer mp/02/JMP [quaternionic parametrizations].
@ 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];
Haber a1912
[relations among the generators in the defining and adjoint representations].
@ 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
= K32
− K12
− K22,
with eigenvalues \(\hbar\)2
k(k−1) (discrete) and \(\hbar\)2
(−λ2−1/4) (continuous);
It 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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