Examples of Lie Groups and Lie Algebras| Examples of Lie Groups and Lie Algebras |
General Linear Groups
* GL(n, R):
The set of n × n invertible real matrices, together with
the operation of ordinary matrix multiplication; It is a non-compact group of
dimension n2; GL(3, \(\mathbb R\))
has orientation-preserving subgroup GL+(3,
\(\mathbb R\)); GL(n, \(\mathbb R\)) = O(n, \(\mathbb R\))
× A(n, \(\mathbb R\)), where A(n, \(\mathbb R\)) is the
set of real, symmetric, positive-definite matrices, which is a contractible space.
* GL(n, C):
The set of n × n invertible complex matrices, together
with the operation of ordinary matrix multiplication; It is a non-compact group
of dimension 2n2.
> Online resources: see
Wikipedia page.
Special Linear Groups > s.a. 4-spinors;
Mandelstam Identities; representations of lie groups.
* SL(n, R):
The set of all n × n real matrices of determinant 1,
together with matrix multiplication; It is a group of dimension
n2−1;
* SL(n, C):
The set of all n × n complex matrices of determinant
1 together with matrix multiplication, a non-compact group of dimension 2
(n2−1); SL(2, \(\mathbb C\))
is sometimes known as the Möbius group.
* Polar decomposition theorem:
Given a 2D complex vector space W and an inner product on W,
any element of SL(2, \(\mathbb C\)) can be written uniquely as the composition
of an element of SU(2) and a positive, self-adjoint map.
Symplectic Groups
* Sp(2n, R):
A non-compact, simple group of dimension n (2n+1),
the subgroup of GL(n, \(\mathbb R\)) which leaves the form
xTJy invariant,
where \(x\), \(y \in \mathbb R^n\), and
\[J = \left(\matrix{0 & {\mathbb 1}_{n\times n}^~ \cr −{\mathbb 1}_{n\times n}^~ &0}\right).\]
* Sp(2): It has the same Lie algebra
as the 2+1 Lorentz group.
> Online resources:
see MathWorld page;
Wikipedia page.
Exceptional Groups > s.a. guts; monopoles
[with group G2]; 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).
@ General references: news BBC(07)mar [E8];
Bernardoni et al ATMP(08)-a0705 [F4, geometry and realization];
Wangberg PhD(07)-a0711 [E6];
Yokota a0902
[simply connected compact simple, elementary introduction];
Borthwick & Garibaldi NAMS-a1012
[on "evidence for E8 symmetry" in the laboratory];
Baez & Huerta TAMS-a1205
[G2 as the symmetry group of a ball rolling on a larger ball];
Cacciatori et al a1207
[explicit generalized Euler angle parameterizations].
@ In physics: Ramond ht/03-conf;
Marrani & Truini a1506-conf [and the nature of spacetime].
Some Relationships
* SL(n, \(\mathbb R\))/SO(n)
= space of symmetric unimodular n × n matrices.
* O(2n, \(\mathbb R\)) ∩
Sp(2n, \(\mathbb R\)) = 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)].
Other Groups > s.a. orthogonal and unitary lie groups
[O(n), SO(p, q), U(n), U(p, q)].
@ With a left-invariant Lorentzian metric:
Calvaruso & Zaeim DG&A(13) [4D];
Anderson & Torre a1911 [spacetime groups].
main page
– abbreviations
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 6 jan 2020