Hyper generalized orthogonal Lie algebra
|
|
- <math>\begin{pmatrix} \mathbf{A} & V \\ -V^t & 0 \end{pmatrix}<math> belongs to <math>\mathfrak{so}<math>(n+1) if A belongs to <math>\mathfrak{so}<math>(n) and V is a (column) n-vector. Skew-symmetric matrix.
- <math>\begin{pmatrix}A & V \\ V^t & 0 \end{pmatrix}<math>
belongs to <math>\mathfrak{so}<math>(n,m+1) if A belongs to <math>\mathfrak{so}<math>(n,m) and V is a (n+m)-vector (including m = 0, of course). Lobachevskian algebra is <math>\mathfrak{so}<math> (n, 1) (not Lorentzian as is usual in the literature, a confusion with his role in Poincare, but neutral is hyperbolic).
New Notation!:<math>\begin{pmatrix}A & V \\ 0 & 0 \end{pmatrix}<math> belongs to <math>\mathfrak{so}<math>(n,m,1) if A belongs to <math>\mathfrak{so}<math>(n,m) and V is a (n+m)-vector. Euclidean algebra is <math>\mathfrak{so}<math>(n,0,1)!. Poincarean algebra is <math>\mathfrak{so}<math>(n,1,1). In general it represents the Lie algebra of the semidirect product of traslations in the space Rn+m with the SO(n, m) which has <math>\mathfrak{so}<math>(n,m) as its Lie algebra.
New Notation:<math>\begin{pmatrix}A & V \\ 0 & 0 \end{pmatrix}<math> belongs to <math>\mathfrak{so}<math>(n,m,l+1) if A belongs to <math>\mathfrak{so}<math>(n,m,l) and V is a (n+m+l)-vector.
In particular:<math>\begin{pmatrix}A & V & X\\ 0 & 0 & t \\0 & 0 & 0\end{pmatrix}<math> belongs to <math>\mathfrak{so}<math>(n,m,2) if A belongs to <math>\mathfrak{so}<math>(n,m) and V and X are (n+m)-vectors. <math>\mathfrak{g}<math>=Galilean algebra is <math>\mathfrak{so}<math>(n,0,2), associated with an iterated semidirect product. (t is a "number", but an important one <math>\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]<math> gives t if n>2. So time is the very conmmutative part of Galilean group).
For completeness we copy here the structure equations dropping nonsense imaginary (in some pages even planckian!) factors. The Lie algebra <math>\mathfrak{g}<math> is spanned by T, Xi, Vi and Aij (antisymmetric tensor) subject to
- [Xi, T] = 0
- [Xi, Xj] = 0
- [Aij, T] = 0
- [Vi, Vj] = 0
- [Aij, Akl] = δik Ajl - δil Ajk - δjk Ail + δjl Aik
- [Aij, Xk] = δik Xj - δjk Xi
- [Aij, Vk] = δik Vj - δjk Vi
- [Vi, Xj] = 0
- [Vi,T]=Xi Oh, is true! velocity is proportion between space and time!
External links
- Geometric Asymptotics by Victor Guillemin and Shlomo Sternberg (Free on AMS) (http://www.ams.org/online_bks/surv14/)
- cf. pg.188 (chIV-7. Periodic Hamiltonian Systems) (http://www.ams.org/online_bks/surv14/surv14-chIV.pdf)