Problems in Geometric Operator Theory (original) (raw)
Mixing and monodromy of abstract polytopes
Transactions of the American Mathematical Society, 2013
The monodromy group Mon(P) of an n-polytope P encodes the combinatorial information needed to construct P. By applying tools such as mixing, a natural group-theoretic operation, we develop various criteria for Mon(P) itself to be the automorphism group of a regular n-polytope R. We examine what this can say about regular covers of P, study a peculiar example of a 4-polytope with infinitely many distinct, minimal regular covers, and then conclude with a brief application of our methods to chiral polytopes.
Reports on Mathematical Physics, 2013
Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension n ≥ −1; now in dim. 2, 3 and 4 there are extra polytopes, while in general dimensions only the hyper-tetrahedron, the hyper-cube and its dual hyper-octahedron exist. We attribute these peculiarites and exceptions to special properties of the orthogonal groups in these dimensions: the SO(2) = U(1) group being (abelian and) divisible, is related to the existence of arbitrarilysided plane regular polygons, and the splitting of the Lie algebra of the O(4) group will be seen responsible for the Schläfli special polytopes in 4-dim., two of which percolate down to three. In spite of dim. 8 being also special (Cartan's triality), we argue why there are no extra polytopes, while it has other consequences: in particular the existence of the three division algebras over the reals R: complex C, quaternions H and octonions O is seen also as another feature of the special properties of corresponding orthogonal groups, and of the spheres of dimension 0,1,3 and 7.
Problems on polytopes, their groups, and realizations
Periodica Mathematica Hungarica, 2006
The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2006.
Polyhedra, lattice structures, and extensions of semigroups
arXiv: Algebraic Geometry, 2020
For an arbitrary rational polyhedron we consider its decompositions into Minkowski summands and, dual to this, the free extensions of the associated pair of semigroups. Being free for the pair of semigroups is equivalent to flatness for the corresponding algebras. Our main result is phrased in this dual setup: the category of free extensions always contains an initial object, which we describe explicitly. These objects seem to be related to unique liftings in log geometry. Further motivation comes from the deformation theory of the associated toric singularity.
An Introduction to Polytope Theory through Ehrhart's Theorem
2019
A classic introduction to polytope theory is presented, serving as the foundation to develop more advanced theoretical tools, namely the algebra of polyhedra and the use of valuations. The main theoretical objective is the construction of the so called Berline-Vergne valuation. Most of the theoretical development is aimed towards this goal. A little survey on Ehrhart positivity is presented, as well as some calculations that lead to conjecture that generalized permutohedra have positive coefficients in their Ehrhart polynomials. Throughout the thesis three different proofs of Ehrhart’s theorem are presented, as an application of the new techniques developed.
Standard Monomial Theory and applications
Representation Theories and Algebraic Geometry, 1998
Notes by Rupert W.T. YU Let V (λ) be the simple g-module of highest weight λ. The aim of this section is to describe an indexing system for a basis of V (λ) of h-eigenvectors. Denote by π λ : t → tλ the path that connects the origin with λ by a straight line.
Presymplectic convexity and (ir)rational polytopes
Journal of Symplectic Geometry, 2019
In this paper, we extend the Atiyah-Guillemin-Sternberg convexity theorem and Delzant's classification of symplectic toric manifolds to presymplectic manifolds. We also define and study the Morita equivalence of presymplectic toric manifolds and of their corresponding framed momentum polytopes, which may be rational or non-rational. Toric orbifolds [15], quasifolds [3] and non-commutative toric varieties [13] may be viewed as the quotient of our presymplectic toric manifolds by the kernel isotropy foliation of the presymplectic form. 3.1. Framed momentum polytopes of presymplectic toric manifolds 10 3.2. Lifting and framing of polytopes 13 3.3. Morita equivalence 15 3.4. Toric orbifolds and quasifolds 18
Trialgebras and families of polytopes
Contemporary mathematics, 2004
We show that the family of standard simplices and the family of Stasheff polytopes are dual to each other in the following sense. The chain modules of the standard simplices, resp. the Stasheff polytopes, assemble to give an operad. We show that these operads are dual of each other in the operadic sense. The main result of this paper is to show that they are both Koszul operads. As a consequence the generating series of the standard simplices and the generating series of the Stasheff polytopes are inverse to each other. The two operads give rise to new types of algebras with 3 generating operations, 11 relations, respectively 7 relations, that we call associative trialgebras and dendriform trialgebras respectively. The free dendriform trialgebra, which is based on planar trees, has an interesting Hopf algebra structure, which will be dealt with in another paper. Similarly the family of cubes gives rise to an operad which happens to be self-dual for Koszul duality. Introduction. We introduce a new type of associative algebras characterized by the fact that the associative product * is the sum of three binary operations : x * y := x ≺ y + x ≻ y + x • y ,