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Ralf Köhl

Publications and source records attributed to Ralf Köhl.

At least 19 recordsLinked to original sources
arXiv2026

A mathematical framework to study organising principles in graphical representations of biochemical processes

The complexity of molecular and cellular processes forces experimental studies to focus on subsystems. To study the functioning of biological systems across levels of structural and functional organisation, we require tools to compose and organise networks with different levels of detail and abstraction. Systems Biology Graphical Notation (SBGN) is a standardised notational system that visualises biochemical processes as networks. Despite their widespread adoption, SBGN languages remain purely visual and lack an underlying mathematical framework, limiting their compositional analysis, abstraction, and integration with formal modelling approaches. SBGN comprises three complementary visual languages-Process Description (SBGN-PD), Activity Flow (SBGN-AF), and Entity Relationship (SBGN-ER)-each operating at a different level of abstraction. In this manuscript, we introduce a category-theoretic formalism for SBGN-PD, a visual language to describe biochemical processes as biochemical reaction networks. Using the theory of structured cospans, we construct a symmetric monoidal double category whose horizontal 1-morphisms correspond to SBGN-PD diagrams. We also analyse how a designated subnetwork influences the surrounding network and how external entities, in turn, affect the internal reactions of the subnetwork. Our work addresses a key gap between biological visualisation and mathematical structure. It provides precise organising principles for SBGN-PD, including compositionality, enabling the construction of large biochemical reaction networks from smaller ones, and zooming out, allowing the abstraction of detailed biochemical mechanisms while preserving their functional interfaces. Throughout the paper, the proposed framework is illustrated using standard SBGN-PD examples, demonstrating its applicability to large-scale biochemical reaction networks.

q-bio.MN↗
arXiv2025

Kac-Moody Symmetric Spaces: arbitrary symmetrizable complex or almost split real type

Kac-Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and the first-named author for centered Kac-Moody groups, that is, Kac-Moody groups that are generated by their root subgroups. In the case of non-invertible generalized Cartan matrices this leads to complications that -- within the approach proposed originally -- cannot be repaired in the affine case. In the present article we propose an alternative approach to Kac-Moody symmetric spaces which for invertible generalized Cartan matrices provides exactly the same concept, which for the non-affine non-invertible case provides alternative Kac-Moody symmetric spaces, and which finally provides Kac-Moody symmetric spaces for affine Kac-Moody groups. In a nutshell, the original intention by Freyn, Hartnick, Horn and Köhl was to construct symmetric spaces that likely lead to primitive actions of the Kac-Moody groups; this, of course, cannot work in the affine case as affine Kac-Moody groups are far from simple. Additionally, we study the Galois descent to almost split real Kac-Moody symmetric spaces based on the theory of almost split Kac-Moody groups developed by Rémy 2002.

math.GR↗
arXiv2025

Computational modelling of bone growth and mineralization surrounding biodegradable Mg-based and permanent Ti implants

In silico testing of implant materials is a research area of high interest, as cost- and labour-intensive experiments may be omitted. However, assessing the tissue-material interaction mathematically and computationally can be very complex, in particular when functional, such as biodegradable, implant materials are investigated. In this work, we expand and refine suitable existing mathematical models of bone growth and magnesium-based implant degradation based on ordinary differential equations. We show that we can simulate the implant degradation, as well as the osseointegration in terms of relative bone volume fraction and changes in bone ultrastructure when applying the model to experimental data from titanium and magnesium-gadolinium implants for healing times up to 32 weeks. An additional sensitivity analysis highlights important parameters and their interactions. Moreover, we show that the model is predictive in terms of relative bone volume fraction with mean absolute errors below 6%.

q-bio.QM↗
arXiv2025

Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras

Given the maximal compact subalgebra $\mathfrak{k}(A)$ of a split-real Kac-Moody algebra $\mathfrak{g}(A)$ of type $A$, we study certain finite-dimensional representations of $\mathfrak{k}(A)$, that do not lift to the maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$ associated to $\mathfrak{g}(A)$ but only to its spin cover $Spin(A)$. Currently, four elementary of these so-called spin representations are known. We study their (ir-)reducibility, semi-simplicity, and lift to the group level. The interaction of these representations with the spin-extended Weyl-group is used to derive a partial parametrization result of the representation matrices by the real roots of $\mathfrak{g}(A)$.

math.RT↗
arXiv2024

Profiniteness of higher rank volume

We show that the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a profinite invariant of octonionic hyperbolic congruence manifolds.

math.GR↗
arXiv2024

On convexity and the Iwasawa decomposition of split real and complex Kac-Moody groups

We prove an analogue of Kostant's convexity theorem for split real and complex Kac-Moody groups associated to free and cofree root data. The result can be seen as a first step towards describing the multiplication map in a Kac-Moody group in terms of Iwasawa coordinates. Our method involves a detailed analysis of the geometry of Weyl group orbits in the Cartan subalgebra of a real Kac-Moody algebra. It provides an alternative proof of Kostant convexity for semisimple Lie groups and also generalizes a linear analogue of Kostant's theorem for Kac-Moody algebras that has been established by Kac and Peterson in 1984.

math.RT↗
arXiv2023

Computing Maxwell eigenmodes with Bloch boundary conditions

Our goal is to predict the band structure of photonic crystals. This task requires us to compute a number of the smallest non-zero eigenvalues of the time-harmonic Maxwell operator depending on the chosen Bloch boundary conditions. We propose to use a block inverse iteration preconditioned with a suitably modified geometric multigrid method. Since we are only interested in non-zero eigenvalues, we eliminate the large null space by combining a lifting operator and a secondary multigrid method. To obtain suitable initial guesses for the iteration, we employ a generalized extrapolation technique based on the minimization of the Rayleigh quotient that significantly reduces the number of iteration steps and allows us to treat families of very large eigenvalue problems efficiently.

math.NA↗
arXiv2022

Topological photonics by breaking the degeneracy of line node singularities in semimetal-like photonic crystals

Degeneracy is an omnipresent phenomenon in various physical systems, which has its roots in the preservation of geometrical symmetry. In electronic and photonic crystal systems, very often this degeneracy can be broken by virtue of strong interactions between photonic modes of the same energy, where the level repulsion and the hybridization between modes causes the emergence of photonic bandgaps. However, most often this phenomenon does not lead to a complete and inverted bandgap formation over the entire Brillouin zone. Here, by systematically breaking the symmetry of a two-dimensional square photonic crystal, we investigate the formation of Dirac points, line node singularities, and inverted bandgaps. The formation of this complete bandgap is due to the level repulsion between degenerate modes along the line nodes of a semimetal-like photonic crystal, over the entire Brillouin zone. Our numerical experimentations are performed by a home-build numerical framework based on a multigrid finite element method. The developed numerical toolbox and our observations pave the way towards designing complete bandgap photonic crystals and exploring the role of symmetry on the optical behaviour of even more complicated orders in photonic crystal systems.

physics.optics↗
arXiv2022

Representations of involutory subalgebras of affine Kac-Moody algebras

We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of $\mathbb{N}$-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity.

math.RT↗
arXiv2021

Fundamental groups of split real Kac-Moody groups and generalized real flag manifolds. With appendices by Tobias Hartnick and Ralf Köhl and by Julius Grüning and Ralf Köhl

We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, the Iwasawa decomposition $G = KAU_+$ provides a weak homotopy equivalence $K \hookrightarrow G$, implying $π_1(G) = π_1(K)$. It thus suffices to determine $π_1(K)$ which we achieve by investigating the fundamental groups of generalized flag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized flag variety is a CW decomposition $-$ in particular, we cover the complete symmetrizable situation; the result concerning the structure of $π_1(K)$ more generally also holds in the non-symmetrizable two-spherical situation.

math.GR↗
arXiv2021

Representations of Quantum Minimal Surface Algebrasvia Kac-Moody-theory

We consider epimorphisms from quantum minimal surface algebras onto involutroy subalgebras of split real simply-laced Kac-Moody algebras and provide examples of affine and finite type. We also provide epimorphisms onto such Kac-Moody algebras themselves, where reality of the construction is important. The results extend to the complex situation.

math.RT↗
arXiv2019

Kac-Moody symmetric spaces

In the present article we introduce and study a class of topological reflection spaces that we call Kac-Moody symmetric spaces. These generalize Riemannian symmetric spaces of non-compact type. We observe that in a non-spherical Kac-Moody symmetric space there exist pairs of points that do not lie on a common geodesic; however, any two points can be connected by a chain of geodesic segments. We moreover classify maximal flats in Kac-Moody symmetric spaces and study their intersection patterns, leading to a classification of global and local automorphisms. Unlike Riemannian symmetric spaces, non-spherical non-affine irreducible Kac-Moody symmetric spaces also admit an invariant causal structure. For causal and anti-causal geodesic rays with respect to this structure we find a notion of asymptoticity, which allows us to define a future and past boundary of such Kac-Moody symmetric space. We show that these boundaries carry a natural polyhedral structure and are cellularly isomorphic to the halves of the geometric realization of the twin buildings of the underlying split real Kac-Moody group. We also show that every automorphism of the symmetric space is uniquely determined by the induced cellular automorphism of the future and past boundary. The invariant causal structure on a non-spherical non-affine irreducible Kac-Moody symmetric space gives rise to an invariant pre-order on the underlying space, and thus to a subsemigroup of the Kac-Moody group. We conclude that while in some aspects Kac-Moody symmetric spaces closely resemble Riemannian symmetric spaces, in other aspects they behave similarly to ordered affine hovels, their non-Archimedean cousins.

math.GR↗
arXiv2018

An extension theorem for embedded Riemannian symmetric spaces of non-compact type and an application to their universal property

It is known that a geodesic Y in an abstract reflection space X in the sense of Loos, without any assumption of differential structure, canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we prove an analog of this result stating that, if X contains an embedded hyperbolic plane H, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL(2,R). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact type embedded in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac-Moody symmetric space G/K for an algebraically simply connected two-spherical Kac-Moody group G satisfies a universal property similar to the universal property that the group G satisfies itself.

math.GR↗
arXiv2017

Extending generalized spin representations

We revisit the construction of higher spin representations by Kleinschmidt and Nicolai for E10, generalize it to arbitrary simply laced types, and provide a coordinate-free approach to the 3/2-spin and 5/2-spin representations. Moreover, we discuss the relationship between our findings and the representation theory of Sym(3) pointed out to us by Levy.

math.RT↗