Hamiltonian Lie algebroids

In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for the Einstein evolution equations of general relativity. The present work was motivated by the effort to explain the coisotropic structure of the constraint subset for the initial value problem by extending the notion of hamiltonian structure from Lie algebra actions to general Lie algebroids over presymplectic manifolds. After comparing possible compatibility conditions between the anchor $A\to TM$ and the presymplectic structure on the base $M$, we choose the most natural of them, given by a suitably chosen connection on $A$. We define a notion of momentum section of $A^*$ and a condition for compatibility with the Lie bracket. A Lie algebroid over a presymplectic manifold with compatible anchor and momentum section is then called hamiltonian. For an action Lie algebroid, we retrieve the conditions of a hamiltonian action. The clean zero locus of the momentum section of a hamiltonian Lie algebroid is a coisotropic submanifold. We show that a bracket-compatible momentum map is equivalent to a closed basic extension of the presymplectic form, within the generalization of the BRST model of equivariant cohomology to Lie algebroids. We construct groupoids by reduction of an action Lie groupoid $G\times M$ by a subgroup $H$ of $G$ which is not necessarily normal, and we find conditions which imply that a hamiltonian structure descends to their Lie algebroids. We consider many examples and, in particular, find that the tangent Lie algebroid over a symplectic manifold is hamiltonian with respect to some connection if and only if the symplectic structure has a nowhere vanishing primitive. Recent results of Stratmann and Tang show that this is the case whenever the symplectic structure is exact

Perturbative Symmetries on Noncommutative Spaces

Perturbative deformations of symmetry structures on noncommutative spaces are studied in view of noncommutative quantum field theories. The rigidity of enveloping algebras of semi-simple Lie algebras with respect to formal deformations is reviewed in the context of star products. It is shown that rigidity of symmetry algebras extends to rigidity of the action of the symmetry on the space. This implies that the noncommutative spaces considered can be realized as star products by particular ordering prescriptions which are compatible with the symmetry. These symmetry preserving ordering prescriptions are calculated for the quantum plane and four-dimensional quantum Euclidean space. Using these ordering prescriptions greatly facilitates the construction of invariant Lagrangians for quantum field theory on noncommutative spaces with a deformed symmetry.Comment: 16 pages; LaTe

Voros product and the Pauli principle at low energies

Using the Voros star product, we investigate the status of the two particle correlation function to study the possible extent to which the previously proposed violation of the Pauli principle may impact at low energies. The results show interesting features which are not present in the computations made using the Moyal star product.Comment: 5 pages LateX, minor correction

q-Deformed Superalgebras

The article deals with q-analogs of the three- and four-dimensional Euclidean superalgebra and the Poincare superalgebra.Comment: 38 pages, LateX, no figures, corrected typo

Braided algebras and the kappa-deformed oscillators

Recently there were presented several proposals how to formulate the binary relations describing kappa-deformed oscillator algebras. In this paper we shall consider multilinear products of kappa-deformed oscillators consistent with the axioms of braided algebras. In general case the braided triple products are quasi-associative and satisfy the hexagon condition depending on the coassociator $Phi \in A\otimes A\otimes A$. We shall consider only the products of kappa-oscillators consistent with co-associative braided algebra, with Phi =1. We shall consider three explicite examples of binary kappa-deformed oscillator algebra relations and describe briefly their multilinear coassociative extensions satisfying the postulates of braided algebras. The third example, describing kappa-deformed oscillators in group manifold approach to kappa-deformed fourmomenta, is a new result.Comment: v2, 13 pages; Proc. of 2-nd Corfu School on Quantum Gravity and Quantum Geometry, September 2009, Corfu; Gen. Rel. Grav. (2011),special Proceedings issue; version in pres

The fundamental pro-groupoid of an affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don't remember \Forget? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the \'etale fundamental groupoid \pi_1(\spec(R)), i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for \'etale \pi_1(\spec(R)) in terms of the category of $R$-modules rather than the category of \'etale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $\pi_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the \'etale fundamental group. For example, \'etale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the \'etale fundamental group of a scheme preserves finite products but not all products.Comment: 46 pages + bibliography. Diagrams drawn in Tik

N=1/2 Deformations of Chiral Superspaces from New Quantum Poincare and Euclidean Superalgebras

We present a large class of supersymmetric classical r-matrices, describing the supertwist deformations of Poincare and Euclidean superalgebras. We consider in detail new family of four supertwists of N=1 Poincare superalgebra and provide as well their Euclidean counterpart. The proposed supertwists are better adjusted to the description of deformed D=4 Euclidean supersymmetries with independent left-chiral and right-chiral supercharges. They lead to new quantum superspaces, obtained by the superextension of twist deformations of spacetime providing Lie-algebraic noncommutativity of space-time coordinates. In the Hopf-algebraic Euclidean SUSY framework the considered supertwist deformations provide an alternative to the N=1/2 SUSY Seiberg's star product deformation scheme.Comment: 17 pages, LaTeX, new (re-worked) revised and extended versio

A Stratification on the Moduli of K3 Surfaces in Positive Characteristic

We review the results on the cycle classes of the strata defined by the height and the Artin invariant on the moduli of K3 surfaces in positive characteristic obtained in joint work with Katsura and Ekedahl. In addition we prove a new irreducibility result for these strata