I investigate contractions via Kac-Moody formalism. In particular, I show how the symmetry algebra of the standard two-dimensional Kepler system, which was identified by Daboul and Slodowy as an infinite-dimensional Kac-Moody loop algebra, and was denoted by ℍ2, gets reduced by the symmetry breaking term, defined by the Hamiltonian H(β) = (1/2m) (p1 2+p22) - α/r- βr-1/2 cos((φ- γ)/2). For this H(β) I define two symmetry loop algebras £i(β), i = 1, 2, by choosing the "basic generators" differently. These £i(β) can be mapped isomorphically onto subalgebras of ℍ2, of codimension two or three, revealing the reduction of symmetry. Both factor algebras £i(β)/ Ii(E, β), relative to the corresponding energy-dependent ideals Ii(E, β), are isomorphic to S-fraktur sign o-fraktur sign(3) and S-fraktur sign o-fraktur sign(2, 1) for E<0 and E>0, respectively, just as for the pure Kepler case. However, they yield two different nonstandard contractions as E→0, namely to the Heisenberg-Weyl algebra h-fraktur sign 3 = r-fraktur sign o-fraktur sign1 or to an Abelian Lie algebra, instead of the Euclidean algebra e-fraktur sign(2) for the pure Kepler case. The above-noted example suggests a general procedure for defining generalized contractions, and also illustrates the "deformation contraction hysteresis," where contraction which involves two contraction parameters can yield different contracted algebras, if the limits are carried out in different order.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics