Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
http://dspace.nbuv.gov.ua:80/handle/123456789/13801
2023-02-06T06:43:36ZMathematical Structure of Loop Quantum Cosmology: Homogeneous Models
http://dspace.nbuv.gov.ua:80/handle/123456789/149374
Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models
Bojowald, M.
The mathematical structure of homogeneous loop quantum cosmology is analyzed, starting with and taking into account the general classification of homogeneous connections not restricted to be Abelian. As a first consequence, it is seen that the usual approach of quantizing Abelian models using spaces of functions on the Bohr compactification of the real line does not capture all properties of homogeneous connections. A new, more general quantization is introduced which applies to non-Abelian models and, in the Abelian case, can be mapped by an isometric, but not unitary, algebra morphism onto common representations making use of the Bohr compactification. Physically, the Bohr compactification of spaces of Abelian connections leads to a degeneracy of edge lengths and representations of holonomies. Lifting this degeneracy, the new quantization gives rise to several dynamical properties, including lattice refinement seen as a direct consequence of state-dependent regularizations of the Hamiltonian constraint of loop quantum gravity. The representation of basic operators - holonomies and fluxes - can be derived from the full theory specialized to lattices. With the new methods of this article, loop quantum cosmology comes closer to the full theory and is in a better position to produce reliable predictions when all quantum effects of the theory are taken into account.
2013-01-01T00:00:00ZRepresentation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
http://dspace.nbuv.gov.ua:80/handle/123456789/149373
Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
de Commer, K.
Let g be a compact simple Lie algebra. We modify the quantized enveloping ∗-algebra associated to g by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping ∗-algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.
2013-01-01T00:00:00ZDirac Operators on Noncommutative Curved Spacetimes
http://dspace.nbuv.gov.ua:80/handle/123456789/149372
Dirac Operators on Noncommutative Curved Spacetimes
Schenkel, A.; Uhlemann, C.F.
We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For general noncommutative curved spacetimes we find that demanding formal self-adjointness as an additional condition singles out a preferred choice among our candidates. Based on this noncommutative Dirac operator we construct a quantum field theory of Dirac fields. In the last part we study noncommutative Dirac operators on deformed Minkowski and AdS spacetimes as explicit examples.
2013-01-01T00:00:00ZA Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
http://dspace.nbuv.gov.ua:80/handle/123456789/149371
A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy
Haine, L.; Vanderstichelen, D.
We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero, Moral and Velázquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] and their action on the tau-functions of the hierarchy is described. The use of the CMV matrices turns out to be crucial for obtaining a Lax pair representation of the master symmetries. The AL hierarchy seems to be the first example of an integrable hierarchy which admits a full centerless Virasoro algebra of master symmetries, in contrast with the Toda lattice and Korteweg-de Vries hierarchies which possess only ''half of'' a Virasoro algebra of master symmetries, as explained in Adler and van Moerbeke [Duke Math. J. 80 (1995), 863-911], Damianou [Lett. Math. Phys. 20 (1990), 101-112] and Magri and Zubelli [Comm. Math. Phys. 141 (1991), 329-351].
2013-01-01T00:00:00Z