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.

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.

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].