Related papers: Reductions related with Hopf maps
Let $M$ be the space of triangles, defined up to shifts, rotations and dilations. We define two maps $f:M\to M$ and $g:M\to M$. The map $f$ corresponds to a triangle of perimeter $\pi$ the triangle with angles numerically equal to edges of…
This paper pursues the study carried out by the authors in "Stability and Hopf bifurcation in a hexagonal governor system", focusing on the codimension one Hopf bifurcations in the hexagonal Watt governor differential system. Here are…
This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of…
This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S-matrix, Feynman…
By studying cohomology classes that are related with $n$-harmonic morphisms and $F$-harmonic maps, we augment and extend several results on $F$-harmonic maps, harmonic maps in [1, 3, 14], $p$-harmonic morphisms in [17], and also revisit our…
For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $\mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive.…
We summarize recent results connecting multiloop Feynman diagram calculations to different parts of mathematics, with special attention given to the Hopf algebra structure of renormalization.
In this survey, we first review some known results on the representation theory of algebras with triangular decomposition, including the classification of the simple modules. We then discuss a recipe to construct Hopf algebras with…
We introduce a new Hopf algebra that operates on pairs of finite interval partitions and permutations of equal length. This algebra captures vincular patterns, which involve specifying both the permutation patterns and the consecutive…
We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:-2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational…
The partition functions of a 2D conformal system - the modular invariant one or the generalized ones, coming from the introduction of defect lines - are expressed in terms of a set of coefficients that have the particularity to form nimreps…
We describe pairs (p,n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of…
Since the Connes--Kreimer Hopf algebra was proposed, revisiting present quantum field theory has become meaningful and important from algebraic points. In this paper, the Hopf algebra in the cutting rules is constructed. Its coproduct…
We give some formulas of the James-Hopf maps by using combinatorial methods. An application is to give a product decomposition of the spaces $\Omega\Sigma^2(X)$.
Two programs, feyngen and feyncop, were developed. feyngen is designed to generate high loop order Feynman graphs for Yang-Mills, QED and $\phi^k$ theories. feyncop can compute the coproduct of these graphs on the underlying Hopf algebra of…
Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor…
The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold…
We investigate Gauss maps associated to great circle fibrations of $S^3$. We show that the associated Gauss map to such a fibration is harmonic (respectively minimal) if and only if the unit vector field generating the great circle…
We determine reductions of 2-dimensional, irreducible, semi-stable, and non-crystalline representations of $\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ with Hodge--Tate weights $0 < k-1$ and with $\mathcal L$-invariant whose $p$-adic…
This paper investigates several distinct attempts to generalize in higher dimension the standard 2-dimensional phyllotaxy set construction. We first recall known contructions for these sets on $2D$ manifolds of constant curvature (the…