ABSTRACT
To compute the law of rjn, we linearize the Lie group Hn. Since the Lie algebra Jjn is associated with the Lie group Hn, we construct a basis B = {eitj | 1 < i < n Az < j < n} of the Lie algebra rjn starting from the one-parameter subgroups <pij : C —> Hn defined as <pij(t) — hn(xrjS(t)), where:
•Er^yt) -it if (r,s) ^ (i,j); if (r,s) = (i,j). (1)
This is due that every one-parameter subgroup on Hn defines an associated 1-dimensional vector space formed by left-invariant differentiate vector fields on Hn and conversely. So, the basis vector field eij is obtained as the associated with the one-parameter subgroup (fij, for 1 < i < n and i < 3 < n-Each basis vector field e ^ can be expressed by:
d dxk,j rr^ dxK3
With respect to this basis B, the law of \)n is determined by the following nonzero brackets: [Ci.h, &h,k\ — Ci fc,
where the subindexes z = 1 , . . . , n, h = i,... ,n and k = h,..., n cannot be equal among themselves at the same time.
