# Hyper generalized orthogonal Lie algebra

[itex]\begin{pmatrix} \mathbf{A} & V \\ -V^t & 0 \end{pmatrix}[itex] belongs to [itex]\mathfrak{so}[itex](n+1) if A belongs to [itex]\mathfrak{so}[itex](n) and V is a (column) n-vector. Skew-symmetric matrix.
[itex]\begin{pmatrix}A & V \\ V^t & 0 \end{pmatrix}[itex]

belongs to [itex]\mathfrak{so}[itex](n,m+1) if A belongs to [itex]\mathfrak{so}[itex](n,m) and V is a (n+m)-vector (including m = 0, of course). Lobachevskian algebra is [itex]\mathfrak{so}[itex] (n, 1) (not Lorentzian as is usual in the literature, a confusion with his role in Poincare, but neutral is hyperbolic).

New Notation!:[itex]\begin{pmatrix}A & V \\ 0 & 0 \end{pmatrix}[itex] belongs to [itex]\mathfrak{so}[itex](n,m,1) if A belongs to [itex]\mathfrak{so}[itex](n,m) and V is a (n+m)-vector. Euclidean algebra is [itex]\mathfrak{so}[itex](n,0,1)!. Poincarean algebra is [itex]\mathfrak{so}[itex](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 [itex]\mathfrak{so}[itex](n,m) as its Lie algebra.

New Notation:[itex]\begin{pmatrix}A & V \\ 0 & 0 \end{pmatrix}[itex] belongs to [itex]\mathfrak{so}[itex](n,m,l+1) if A belongs to [itex]\mathfrak{so}[itex](n,m,l) and V is a (n+m+l)-vector.

In particular:[itex]\begin{pmatrix}A & V & X\\ 0 & 0 & t \\0 & 0 & 0\end{pmatrix}[itex] belongs to [itex]\mathfrak{so}[itex](n,m,2) if A belongs to [itex]\mathfrak{so}[itex](n,m) and V and X are (n+m)-vectors. [itex]\mathfrak{g}[itex]=Galilean algebra is [itex]\mathfrak{so}[itex](n,0,2), associated with an iterated semidirect product. (t is a "number", but an important one [itex]\mathfrak{g}/[\mathfrak{g},\mathfrak{g}][itex] 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 [itex]\mathfrak{g}[itex] 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!

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