Solvable Lie algebras with naturally graded nilradicals and their invariants

Abstract

The indecomposable solvable Lie algebras with graded nilradical of maximal nilindex and a Heisenberg subalgebra of codimension one are analyzed, and their generalized Casimir invariants calculated. It is shown that rank one solvable algebras have a contact form, which implies the existence of an associated dynamical system. Moreover, due to the structure of the quadratic Casimir operator of the nilradical, these algebras contain a maximal non-abelian quasi-classical Lie algebra of dimension 2n − 1, indicating that gauge theories (with ghosts) are possible on these subalgebras.

Solvable extensions of a special class of nilpotent Lie algebras

“…It would be interesting to compare the results obtained in this article and elsewhere with the idea of naturally graded algebras introduced in [1,2,3]. In particular Campoamor-Stursberg and his colleagues find that in order for an algebra to be naturally graded it must be filiform although they do not use this terminology.…”

“…There is only one solvable indecomposable Lie algebra up to isomorphism with a codimension two nilradical denoted g n+2, 1 . …”

“…Therefore to find solvable algebras we need to simplify D 1 as much as possible by changing basis such that the brackets of N n,11 are unchanged. There are a number of cases depending on the possible values of a 1 and b 1 . Although we state a number of conditions at the beginning of each case they only become apparent after the fact as we work through the calculation and simplify the form of the algebra.…”

“…Further we apply the transformation which fixes everything but e 1 such that e 1 [e n , e n+1 ] = (7 − 2(n − i))e n , (1 ≤ i ≤ n − 5, n ≥ 6) .…”

Invariants of triangular Lie algebras

“…The actual problem is the investigation of generalized Casimir operators for classes of solvable Lie algebras or non-solvable Lie algebras with non-trivial radicals of arbitrary finite dimension. There are a number of papers on the partial classification of such algebras and the subsequent calculation of their invariants [1,6,7,14,15,16,20,21,22,23]. In particular, Tremblay and Winternitz [22] classified all the solvable Lie algebras with the nilradicals isomorphic to the nilpotent algebra t 0 (n) of strictly upper triangular matrices for any fixed dimension n. Then in [23] invariants of these algebras were considered.…”