Solvable Lie algebras with naturally graded nilradicals and their invariants

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Solvable Lie algebras with naturally graded nilradicals and their invariants
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2006 J. Phys. A: Math. Gen. 39 1339
(http://iopscience.iop.org/0305-4470/39/6/008)
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I
NSTITUTE OF
P
HYSICS
P
UBLISHING
J
OURNAL OF
P
HYSICS
A: M
ATHEMATICAL AND
G
ENERAL
J. Phys. A: Math. Gen.
39
(2006) 1339–1355
doi:10.1088/0305-4470/39/6/008
Solvable Lie algebras with naturally graded
nilradicals and their invariants
J M Ancochea, R Campoamor-Stursberg and L Garcia Vergnolle
Departamento Geometr´ıa y Topolog´ıa, Fac. CC. Matem´aticas, Universidad Complutense de
Madrid, Plaza de Ciencias 3, E-28040 Madrid, Spain
E-mail:
[e-mail]
,
[e-mail]
and
[e-mail]
Received 17 October 2005, in final form 19 December 2005
Published 25 January 2006
Online at
stacks.iop.org/JPhysA/39/1339
Abstract
The indecomposable solvable Lie algebras with graded nilradical of maximal
nilindex and a Heisenberg subalgebra of codimension one are analysed, and
their generalized Casimir invariants are 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 2
n
−
1, indicating that gauge theories
(with ghosts) are possible on these subalgebras.
PACS numbers: 02.20.Sv, 02.20.Qs, 03.65.Fd