Generalized orthogonal Lie algebra

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A generalized orthogonal Lie algebra is a Lie algebra associated with a generalized orthogonal group. This type of algebras are characterized by two integers n and m so that for every pair of positive integers with m. n has a type of generalized orthogonal algebra, algebra: ${displaystyle {mathfrak {so}}(n,m)}$ . The vectorial dimension of these Lie algebras is given by:

${displaystyle dim({mathfrak {so}}(n,m))={begin{pmatrix}n\2end{pmatrix}}}$

A second generalization discussed in this article explains how to build algebras called ${displaystyle {mathfrak {so}}(n,m)}$ from the previous ones.

Representation and examples

Algebras so(n,m)

There is an obvious way to represent Lie algebras for values of n and m in terms of lower-dimensional algebras. For example:

${displaystyle {begin{pmatrix}mathbf {A} &V\-V^{t}&0end{pmatrix}}}$ belongs to ${displaystyle {mathfrak {so}}}$ (n+1) yes A belongs to ${displaystyle {mathfrak {so}}}$ (n) and V It's a n-vector (columna).
${displaystyle {begin{pmatrix}mathbf {A} &V\V^{t}&0end{pmatrix}}}$ belongs to ${displaystyle {mathfrak {so}}}$ (n, m+1) yes A belongs to ${displaystyle {mathfrak {so}}}$ (n, m) and V is a (n+m)-vector.(including m = 0, of course). The algebra of Lobachevski is ${displaystyle {mathfrak {so}}}$ (n,1) (not algebra of Lorentz as usual in literature, a confusion with its role in the algebra of Poincaré, although the common expression is hyperbolic algebra).

Algebras so(n,m,l)

The above construction can be generalized a bit further by the following notation:

{displaystyle {begin{pmatrix}A&V\0&0end{pmatrix}}}

belongs to

{displaystyle {mathfrak {so}}}

(n, m(1) if A belongs to

{displaystyle {mathfrak {so}}}

(n, m) and V is a (n+m)-vector. The euclidian algebra is

{displaystyle {mathfrak {so}}}

(n0, 1)! The algebra of Poincaré is

{displaystyle {mathfrak {so}}}

(n1:1. In general, it represents the Lie algebra of the semi-direct product of the translations in space R^n+m with SO(n, m) that has

{displaystyle {mathfrak {so}}}

(n, mLike his algebra of Lie.

New Note: ${displaystyle {begin{pmatrix}A&V\0&0end{pmatrix}}}$ belongs to ${displaystyle {mathfrak {so}}}$ (n, m, l+1) yes A belongs to ${displaystyle {mathfrak {so}}}$ (n, m, l) and V is a (n+m+l)-vector.

In particular: ${displaystyle {begin{pmatrix}A&V&X\0&0&t\0&0&0end{pmatrix}}}$ belongs to ${displaystyle {mathfrak {so}}}$ (n, m,2) yes A belongs to ${displaystyle {mathfrak {so}}}$ (n, m) and V and X are (n+mV.P. Galileo's algebra is ${displaystyle {mathfrak {so}}}$ (n,0,2), associated with an iterated semi-direct product. (t It's a "number," but important. ${displaystyle {mathfrak {g}}/[{mathfrak {g}},{mathfrak {g}}]}$ da t Yeah. n2. So time is the commutative part of the Galileo group.

To complete, we give here the equations of structure. The Algebra of Galileo ${mathfrak {g}}$ is expanded by T, X_i, V_i and A_ij (antisymmetric intensity) according to:

[X_i, T] = 0
[X_i, X_j] = 0
[A_ij, T] = 0
[VV_i, V_j] = 0
[A_ijA_kl] = δ_ik A_jl - δ_il A_jk - δ_jk A_il + δ_jl A_ik
[A_ij, X_k] = δ_ik X_j - δ_jk X_i
[A_ij, V_k] = δ_ik V_j - δ_jk V_i
[VV_i, X_j] = 0
[VV_i,T]=X_i

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