Spin Representations of the q-Poincare Algebra

The spin of particles on a non-commutative geometry is investigated within the framework of the representation theory of the q-deformed Poincare algebra. An overview of the q-Lorentz algebra is given, including its representation theory with explicit formulas for the q-Clebsch-Gordan coefficients. The vectorial form of the q-Lorentz algebra (Wess), the quantum double form (Woronowicz), and the dual of the q-Lorentz group (Majid) are shown to be essentially isomorphic. The construction of q-Minkowski space and the q-Poincare algebra is reviewed. The q-Euclidean sub-algebra, generated by rotations and translations, is studied in detail. The results allow for the construction of the q-Pauli-Lubanski vector, which, in turn, is used to determine the q-spin Casimir and the q-little algebras for both the massive and the massless case. Irreducible spin representations of the q-Poincare algebra are constructed in an angular momentum basis, accessible to physical interpretation. It is shown how representations can be constructed, alternatively, by the method of induction. Reducible representations by q-Lorentz spinor wave functions are considered. Wave equations on these spaces are found, demanding that the spaces of solutions reproduce the irreducible representations. As generic examples the q-Dirac equation and the q-Maxwell equations are computed explicitly and their uniqueness is shown.Comment: Submitted as Ph.D. Thesis on March 8, 2001. Ph.D thesis, Ludwig-Maximilians-Universitaet Muenchen, 200

Removable presymplectic singularities and the local splitting of Dirac structures

We call a singularity of a presymplectic form $\omega$ removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinors. All removable singularities are poles in the sense that the norm of $\omega$ is not locally bounded. The points at which removable singularities occur are the non-regular points of the Dirac structure for which we prove a general splitting theorem: Locally, every Dirac structure is the gauge transform of the product of a tangent bundle and the graph of a Poisson structure. This implies that in a neighborhood of a removable singularity $\omega$ can be split into a non-singular presymplectic form and a singular presymplectic form which is the partial inverse of a Poisson bivector that vanishes at the singularity. An interesting class of examples is given by log-Dirac structures which generalize log-symplectic structures. The analogous notion of removable singularities of Poisson structures is also studied.Comment: 23 pages, improvements from referee report

Spin in the q-Deformed Poincare Algebra

We investigate spin as algebraic structure within the q-deformed Poincare algebra, proceeding in the same manner as in the undeformed case. The q-Pauli-Lubanski vector, the q-spin Casimir, and the q-little algebras for the massless and the massive case are constructed explicitly

The homotopy momentum map of general relativity

We show that the action of spacetime vector fields on the variational bicomplex of general relativity has a homotopy momentum map that extends the map from vector fields to conserved currents given by Noether's first theorem to a morphism of $L_\infty$-algebras.Comment: 27 pages, typos fixe

Stacky Lie groups

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2-category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As example we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus.Comment: 40 pages; definition of group objects in higher categories added; coherence relations for groups in 2-categories given (section 4

Hamiltonian Lie algebroids over Poisson manifolds

We extend to Poisson manifolds the theory of hamiltonian Lie algebroids originally developed by two of the authors for presymplectic manifolds. As in the presymplectic case, our definition, involving a vector bundle connection on the Lie algebroid, reduces to the definition of hamiltonian action for an action Lie algebroid with the trivial connection. The clean zero locus of the momentum section of a hamiltonian Lie algebroid is an invariant coisotropic submanifold, the distribution being given by the image of the anchor. We study some basic examples: bundles of Lie algebras with zero anchor and cotangent and tangent Lie algebroids. Finally, we discuss a suggestion by Alejandro Cabrera that the conditions for a Lie algebroid $A$ to be hamiltonian may be expressed in terms of two bivector fields on $A^*$, the natural Poisson structure on the dual of a Lie algebroid and the horizontal lift by the connection of the given Poisson structure on the base.Comment: 29 pages, added section on Poisson reduction, typos fixe