Abstract
We show how the Newton–Hooke (NH) symmetries, representing a nonrelativistic version of de-Sitter symmetries, can be enlarged by a pair of translation vectors describing in Galilean limit the class of accelerations linear in time. We study the Cartan–Maurer one-forms corresponding to such enlarged NH symmetry group and by using cohomological methods we determine the general 2-parameter (in D=2+1 4-parameter) central extension of the corresponding Lie algebra. We derive by using nonlinear realizations method the most general group—invariant particle dynamics depending on two (in D=2+1 on four) central charges occurring as the Lagrangean parameters. Due to the presence of gauge invariances we show that for the enlarged NH symmetries quasicovariant dynamics reduces to the one following from standard NH symmetries, with one central charge in arbitrary dimension D and with second exotic central charge in D=2+1.
Highlights
In the classification of all kinematical groups in D = 3 + 1 one finds the two nonrelativistic counterparts of dS and AdS symmetries - two Newton-Hooke (NH) cosmological groups generated by two NH nonrelativistic algebras [1, 2, 3, 4]
We study the Cartan-Maurer one-forms corresponding to such enlarged NH symmetry group and by using cohomological methods we determine the general 2-parameter central extension of the corresponding Lie algebra
We derive by using nonlinear realizations method the most general group - invariant particle dynamics depending on two central charges occurring as the Lagrangean parameters
Summary
In arbitrary dimension D one can introduce one central charge described by the mass parameter m [Pi, Kj] = m δij. We denote corresponding algebra by NH and list below the relations satisfied by new generators Fi, in arbitrary dimension D with central extensions included:. We see that the relation (1.4d) does not provide a finite limit R → ∞; in order to obtain such an Galilean limit we should introduce R-dependent Newton-Hooke mass parameter m. The parameters appearing in the lagrangean are (besides the dS radius R) in one to one correspondence with the central extensions of the NH algebra. The plan of the paper is as follows: in section 2 we will introduce the double enlarged NH algebra and its central extension; the case D = 2 + 1.
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