Quasi-invariant Lagrangians on Lie groups and the method of coadjoint orbits

Abstract

Starting with a two-cocycle globally defined on a Lie group G, the Lagrangian system on G is constructed whose Lagrangian is quasi-invariant under a right translation canonically lifted to the tangent bundle TG. A direct consequence of the quasi-invariance is the appearance of central extensions in the Noether-symmetry algebra. Then, it is shown that the kinematical sector of such a model realizes, through a symplectic reduction, the Kirillov–Kostant symplectic structure on the coadjoint orbit of the Lie group with the central extensions. The generalization to a ‘‘higher-dimensional’’ theory of the Wess–Zumino–Witten-type is also discussed herein.

Graded contractions of Lie algebras and central extensions*

“…as a set of solutions of (11). The equations (12) and 13are the central result of this paper. Before solving (13), one should note that they must be slightly modified in three special circumstances:…”

“…, and (iii) the charges β (µ,i),(ν,j) are forced to be 0 for all i, j. Under such circumstances, the relevant term in equation 11does not contribute to the sum and any product containing ε µ,ν or η µ,ν must be taken out of the relations (12) and (13). The parameters ε µ,ν or η µ,ν are then referred to as irrelevant.…”

“…3. Contract L to L by first removing the appropriate terms in (12) when the grading is nongeneric, and then solving the equations for the ε's;…”

“…The importance of central charges in physics is likewise well appreciated [8]- [12]. Their appearance in a Lie algebra can be the counterpart of the presence of non-trivial phases in a projective representation of the corresponding group.…”