Examples
of Lie Groups and Lie Algebras| Examples
of Lie Groups and Lie Algebras |
General Linear Groups
* GL(n, R):
The
dimension is n2; Non-compact; GL(3, R)
has orientation-preserving subgroup GL+(3, R);
GL(n,R) = O(n,R) 
A(n,R),
where A(n,R) is the set of real, symmetric, positive-definite
matrices, which is a contractible space.
* GL(n, C):
The
dimension is 2n2; Non-compact.
Special Linear Groups > s.a. Mandelstam
Identities.
* SL(n, R):
The
dimension is n2–1;
* SL(n, C):
The
dimension is 2 (n2–1); Non-compact.
* Polar decomposition theorem:
Given a 2D complex vector space W and
an inner product on W, any element of SL(2, C) can
be written uniquely as
the composition of an element of SU(2) and a positive, self-adjoint map.
Orthogonal Groups > s.a. fundamental
groups; guts [SO(10)].
* O(n): The
dimension is n (n–1)/2.
* SO(n): The
dimension is n (n–1)/2.
* SO(3): 
RP3,
locally the same as SU(2), simple.
* SO(4): Isomorphic to
(SU(2) ×
SU(2))/Z2 [@
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 
R4,
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.
@ O(n): Gorin JMP(02)mp/01,
Braun JPA(06)mp [integrals,
> s.a. lie groups].
@ SO(n): Alisauskas
JPA(02)mp [3j symbols],
JPA(02)
[6j symbols]; Jiang & Soudry AM(03)
[local
converse theorem for SO(2n+1)]; Mukunda et al a0904 [SO(3),
Hamilton's
theory of turns]; > s.a. SU(2).
Pseudo-Orthogonal Groups > s.a. fundamental
groups; lorentz
group [SO(3,1)]; Racah Coefficients.
* O(p, q):
* SO(p, q):
Noncompact; SO(2,1)
= SL(2,R)/Z2;
SO(3,1) is simple; SO(2,2) = SL(2,R) × SL(2,R).
@ References: Alhaidari PRA(02)mp/01 [SO(2,1),
graded extension and physics].
Unitary Groups > s.a. holonomy [U(1)]; lie
algebra; representations;
standard model; SU(2).
* 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(infty)
and SU(infty): Inductive
limits of U(n) and SU(n),
respectively.
@ SU(3): Gsponer mp/02/JMP
[quaternionic parametriz]; Kerner a0901 [from Z3-
graded cubic algebra]; Shurtleff a0908 [formulas for matrixes].
@ 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].
@ U(n):
Tilma & Sudarshan JGP(04)mp/02 [Euler
angles]; Aubert & Lam JMP(03)mp,
JMP(04)mp [integration].
@ U(infty) and SU(infty):
in Mavromatos & Winstanley CQG(00)ht/99;
Borodin & Olshanski AM(05)m.RT/01 [harmonic
analysis]; Swain ht/04,
ht/04, ht/04 [SU(
)
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
algebra.
* U(p, q):
* SU(p, q):
Non-compact.
* SU(1, 1): 3D; Casimir
invariant
C2 = K32–K12–K22,
with eigenvalues 
2k(k–1)
(discrete) and 2(–
2–1/4)
(continuous); Can be parametrized by 
,
C
with ||2 +
||2 =
1, for example as
Symplectic Groups
* Sp(2n, R):
A non-compact, simple group of dimension n (2n–1),
the subgroup of GL(n, R)
which leaves the form xTJy invariant,
where x,
y Rn,
and J is the matrix {0 & I \cr –I &0}.
* Sp(2):
Same Lie algebra
as the 2+1 Lorentz group.
Exceptional Groups > s.a. guts; unified
theories.
* Finite ones: G2 (14-dimensional);
F4 (52-dimensional); E6 (78-dimensional);
E7 (133-dimensional) and E8 (248-dimensional,
the largest simple exceptional Lie group).
@ References: Ramond ht/03-in
[in physics]; news BBC(07)mar
[E8]; Bernardoni et al ATMP(08)-a0705 [F4,
geometry and realization]; Wangberg a0711-PhD
[E6].
Some Relationships
* SL(n, R)/SO(n)
= space of symmetric unimodular n × n matrices.
* O(2n, R) 
Sp(2n,R)
= U(n).
* SU(2)/U(1) = S2, not a group, since U(1) is not a normal subgroup.
* SO(3)/SO(2) = S2, not a group, since SO(2) is not a normal subgroup.
@ References: Fujii et al IJGMP(07)qp/06
[SU(2) 
SU(2)
= SO(4)].
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