SOME LOOP ALGEBRAS OF HAMILTONIAN LIE ALGEBRAS

We consider a class of thin Lie algebras with second diamond in weight a power of the characteristic of the underlying field. We identify these Lie algebras with loop algebras of a graded Hamiltonian algebra or loop algebras of an extension of the Hamiltonian algebra by an outer derivation. We also prove that the Lie algebras considered are not finitely presented.

ITERATED PATH INTEGRALS REALIZATION OF QUANTUM VORTEX CURRENTS: CONSTRUCTION OF THE TOPOLOGICAL INVARIANTS

Resorting to the notions of iterated path integrals and generalized connections which appear in K.T. Chen’s theory, it is shown how the current algebra arising in the framework of the Rasetti and Regge theory of quantum vortices—that is known to coincide with the hamiltonian algebra pertaining to a certain coadjoint orbit of the group of measure-preserving diffeomorphisms of the fluid ambient space—allows the complete reconstruction of the topology of the link supporting the vorticity field. It is thus proved the conjecture that the set of topological invariants for such link, which are represented by the central elements of the subgroups of the lower central series for its fundamental group, are among the Casimir operators of the current algebra, i.e. the constants of motion of the vortex system.