Deformation and Contraction of Symmetries in Special Relativity

This dissertation gives an account of the fundamental principles underlying two conceptionally different ways of embedding Special Relativity into a wider context. Both of them root in the attempt to explore the full scope of the Relativity Postulate. The first approach uses Lie algebraic analysis alone, but already yields a whole range of alternative kinematics that are all in a quantifiable sense near to those in Special Relativity, while being rather far away in a qualitative way. The corresponding models for spacetime are seen to be four-dimensional versions of the prototypical planar geometries associated with the work of Cayley and Klein. The close relationship between algebraic and geometric methods displayed by these considerations is being substantialized in terms of light-like spacetime extensions. The second direction of departures from Special Relativity stresses and develops the algebraic view on spacetime by considering Hopf instead of Lie algebras as candidates for the description of kinematical transformations and hence spacetime symmetry. This approach is motivated by the belief in the existence of a quantum theory of gravity, and the assumption that such manifests itself in nonlinear modifications of the laws of Special Relativity at length scales comparable to the Planck length. The twofold character of this work, and the presentation of an example for the fully geometric character of a specific Hopf algebraic deformation of the PoincareI algebra, enable a conclusion that speculates on a possible relationship between the two developed viewpoints via the technique of nonlinear realizations. A non-perturbative approach to the latter is given which generalizes to all the considered geometries

The Many Classical Faces of Quantum Structures

Interpretational problems with quantum mechanics can be phrased precisely by only talking about empirically accessible information. This prompts a mathematical reformulation of quantum mechanics in terms of classical mechanics. We survey this programme in terms of algebraic quantum theory.Comment: 24 page

Galois theory in monoidal categories

The Galois theory of Chase and Sweedler [11], for commutative rings, is generalized to encompass commutative monoids in an arbitrary symmetric, closed, monoidal category with finite limits and colimits. The primary tool is the Morita theory of Pareigis [35, 36, 37], which also supplies a suitable definition for the concept of a “finite” object in a monoidal category. The Galois theory is then extended by an examination of “normal” sub-Hopf-monoids, and examples in various algebraic and topological categories are considered. In particular, symmetric, closed, monoidal structures on various categories of topological vector spaces are studied with respect to the existence of “finite” objects

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

This Week's Finds in Mathematical Physics (1-50)

These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of $n$-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird).Comment: 242 page