18 research outputs found
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 and the presymplectic structure on the base , we choose the most natural of them, given by a suitably chosen connection on . We define a notion of momentum section of 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 by a subgroup of 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 . 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 , 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 when it is a field. This gives a new definition
for \'etale \pi_1(\spec(R)) in terms of the category of -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 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