Examples of Lie Groups and Lie Algebras |

**General Linear Groups**

* __GL( n, R)__:
The set of

*

>

**Special Linear Groups** > s.a. 4-spinors;
Mandelstam Identities; representations of lie groups.

* __SL( n, R)__:
The set of all

*

*

**Symplectic Groups**

* __Sp(2 n, R)__:
A non-compact, simple group of dimension

\[J = \left(\matrix{0 & {\mathbb 1}_{n\times n}^~ \cr −{\mathbb 1}_{n\times n}^~ &0}\right).\]

* __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 G_{2}]; unified theories.

* __Finite ones__: G_{2} (14-dimensional);
F_{4} (52-dimensional); E_{6} (78-dimensional);
E_{7} (133-dimensional) and E_{8} (248-dimensional,
the largest simple exceptional Lie group).

@ __General references__: news BBC(07)mar [E_{8}];
Bernardoni et al ATMP(08)-a0705 [F_{4}, geometry and realization];
Wangberg PhD(07)-a0711 [E_{6}];
Yokota a0902
[simply connected compact simple, elementary introduction];
Borthwick & Garibaldi NAMS-a1012
[on "evidence for E8 symmetry" in the laboratory];
Baez & Huerta TAMS-a1205
[G_{2} 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(2*n*, \(\mathbb R\)) ∩
Sp(2*n*, \(\mathbb R\)) = U(*n*).

* SU(2)/U(1) = S^{2},
not a group, since U(1) is not a normal subgroup.

* SO(3)/SO(2) = S^{2},
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*)].

@ __References__: Calvaruso & Zaeim DG&A(13) [4D Lie groups with a left-invariant Lorentzian metric].

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send feedback and suggestions to bombelli at olemiss.edu – modified 26 aug 2018