Bas Janssens

Characterization of Reflection Positivity: Majoranas and Spins

We study linear functionals on a Clifford algebra (algebra of Majoranas) equipped with a reflection automorphism. For Hamiltonians that are functions of Majoranas or of spins, we find necessary and sufficient conditions on the coupling constants for reflection positivity to hold. One can easily check these conditions in concrete models. We illustrate this by discussing a number of spin systems with nearest-neighbor and long-range interactions.

Universal Central Extensions of Gauge Algebras and Groups

We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.

Universal Central Extension of the Lie Algebra of Hamiltonian Vector Fields

We determine the universal central extension of the Lie algebra of hamiltonian vector fields, thereby classifying its central extensions. Furthermore, we classify the central extensions of the Lie algebra of symplectic vector fields, of the Poisson Lie algebra, and of its compactly supported version.

Momentum Maps for Smooth Projective Unitary Representations

For a smooth projective unitary representation (ρ,H) of a locally convex Lie group G, the projective space P(H∞) of smooth vectors is a locally convex Kahler manifold. We show that the action of G on P(H∞) is weakly Hamiltonian, and lifts to a Hamiltonian action of the central U(1)- extension G # obtained from the projective representation. We identify the non-equivariance cocycles obtained from the weakly Hamiltonian action with those obtained from the projective representation, and give some integrality conditions on the image of the momentum map.