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Hans Cuypers

Publications and source records attributed to Hans Cuypers.

13 recordsLinked to original sources
arXiv2026

On Orthogonal Graphs and their Automorphisms

We provide a characterization of the connected subgraphs of the graphs with vertex set the non-isotropic points in a quadratic space $(V,Q)$, two points adjacent if and only if they span a tangent line. Here $(V,Q)$ is a quadratic space $V$ over a finite field $\mathbb{F}_q$ of order $q$, where $q>3$ is odd, equipped with a non-degenerate quadratic form $Q$. The local structure of the graph (i.e. the graph induced on the neighbors of a point) determines the structure of the full graph. This characterization helps to determine the automorphism group of these graphs.

math.CO↗
arXiv2026

Quasi-Clifford algebras, Quadratic forms over $\mathbb{F}_2$, and Lie Algebras

Let $Γ=(\mathcal{V},\mathcal{E})$ be a graph, whose vertices $v\in \mathcal{V}$ are colored black and white and labeled with invertible elements $λ_v$ from a commutative and associative ring $R$ containing $\pm 1$. Then we consider the associative algebra $\mathfrak{C}(Γ)$ with identity element $\mathbf{1}$ generated by the elements of $\mathcal{V}$ such that for all $v,w\in \mathcal{V}$ we have \[\begin{array}{lll}v^2 &=λ_v\mathbf{1}&\textrm{if } v \textrm{ is white}, v^2 &=-λ_v\mathbf{1}&\textrm{if } v \textrm{ is black}, vw+wv&=0&\textrm{if } \{v,w\}\in \mathcal{E}, vw-wv&=0&\textrm{if } \{v,w\}\not\in \mathcal{E}.\\ \end{array}\] If $Γ$ is the complete graph, $\mathfrak{C}(Γ)$ is a Clifford algebra, otherwise it is a so-called quasi-Clifford algebra. We describe this algebra as a twisted group algebra with the help of a quadratic space $(V,Q)$ over the field $\mathbb{F}_2$. Using this description, we determine the isomorphism type of $\mathfrak{C}(Γ)$ in several interesting examples. As the algebra $\mathfrak{C}(Γ)$ is associative, we can also consider the corresponding Lie algebra and some of its subalgebras. In case $λ_v=1$ for all $v\in \mathcal{V}$, and all vertices are black, we find that the elements $v,w\in \mathcal{V}$ satisfy the following relations $$\begin{array}{lll} [v,w]&=0&\textrm{if } \{v,w\}\not\in \mathcal{E}, {[v,[v,w]]}&=-w&\textrm{if } \{v,w\}\in \mathcal{E}.\\ \end{array}$$ In case $R$ is a field of characteristic $0$, we identify these algebras as quotients of the compact subalgebras of Kac-Moody Lie algebras and prove that they admit a so-called generalized spin representation.

math.RA↗
arXiv2026

Dynamical Lie algebras generated by Pauli strings and quadratic spaces over $\mathbb{F}_2$

Dynamical Lie algebras, i.e. Lie subalgebras of $\mathfrak{su}(2^n)$, generated by Pauli strings have recently been studied intensively. They are also called Pauli Lie algebras or Hamiltonian Lie algebras. In this paper we provide a uniform mathematical approach to various recent results on Pauli Lie algebras. Moreover, we present an algorithm that on input of a set of Pauli strings determines the isomorphism type of the dynamical Lie algebra generated by these Pauli's in time $\mathcal{O}(\max(n,m)^3)$ where $m$ is the size of the generating set.

quant-ph↗
arXiv2026

Graphs, Axial Algebras and their Automorphism Groups

We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We determine their fusion laws, prove them to be simple in almost all cases, and determine their automorphism group under some conditions on the degrees and girth of the graph. A construction of a class of these graphs with prescribed automorphism group enables us to construct for each group $G$ infinitely many simple (axial) algebras (with a fixed fusion law) such that the automorphism group of the algebra is isomorphic to $G$.

math.AC↗
arXiv2022

Characterizations of symplectic polar spaces

A polar space S is said to be symplectic if it admits an embedding e in a projective geometry PG(V) such that the e-image e(S) of S is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when S admits different (non isomorphic) embeddings, as it is the case (precisely) when S is defined over a field of characteristic 2.

math.SG↗
arXiv2021

A geometric characterization of the finitary special linear and unitary Lie algebras

An extremal element $x$ in a Lie algebra $\mathfrak{g}$ is an element for which the space $[x, [x, \mathfrak{g}]]$ is contained in the linear span of $x$. Long root elements in classical Lie algebras are examples of extremal elements. Lie algebras generated by extremal elements lead to geometries with as points the $1$-spaces generate by extremal elements an as lines the $2$-spaces whose non-zero elements are pairwise commuting extremal elements. In this paper we show that the finitary special linear Lie algebras can be characterized by their extremal geometry. Moreover, we also show that the finitary special unitary Lie algebras can be characterized by the fact that their geometry has no lines, but that after extending the field quadratically, the geometry is that of a special linear Lie algebra.

math.RA↗
arXiv2021

Whitney's Theorem for Line Graphs of Multi-Graphs

Whitney's Theorem states that every graph, different from $K_3$ or $K_{1,3}$, is uniquely determined by its line graph. A $1$-line graph of a multi-graph is the graph with as vertices the edges of the multi-graph, and two edges adjacent if and only if there is a unique vertex on both edges. The $\geq 1$-line graph of a multi-graph is the graph on the edges of the multi-graph, where two edges are adjacent if and only if there is at least one vertex on both edges. We extend Whitney's theorem to such line graphs of multi-graphs, and show that most multi-graphs are uniquely determined by their line graph. Moreover, we present an algorithm to determine for a given graph $Γ$, if possible, a multi-graph with $Γ$ as line graph.

math.CO↗
arXiv2021

Line graphs of Multi-Graphs and the forbidden graph $E_6$

The line graph $Γ$ of a multi-graph $Δ$ is the graph whose vertices are the edges of $Δ$, where two such edges are adjacent if and only if they meet in a single vertex of $Δ$. We provide several characterizations of such line graphs and in particular show that a graph is a line graph if and only if it does not contain one of $33$ graphs, all of which correspond to bases of anisotropic vectors of a $6$-dimensional orthogonal geometry of $-$-type over a field with two elements, or, equivalently, to sets of $6$ generating reflections in the Weyl group of type $E_6$.

math.CO↗
arXiv2017

A geometric characterization of the classical Lie algebras

A nonzero element x in a Lie algebra g over a field F with Lie product [ , ] is called a extremal element if [x, [x, g]] is contained in Fx. Long root elements in classical Lie algebras are examples of extremal elements. Arjeh Cohen et al. initiated the investigation of Lie algebras generated by extremal elements in order to provide a geometric characterization of the classical Lie algebras generated by their long root el- ements. He and Gabor Ivanyos studied the so-called extremal geometry with as points the 1-dimensional subspaces of g generated by extremal elements of g and as lines the 2-dimensional subspaces of g all whose nonzero vectors are extremal. For simple finite dimensional g this geometry turns out to be a root shadow space of a spherical building. In this paper we show that the isomorphism type of g is determined by its extremal geometry, provided the building has rank at least 3.

math.RA↗
arXiv2017

A geometric characterization of the symplectic Lie algebra

A nonzero element $x$ in a Lie algebra $\mathfrak{g}$ with Lie product $[ , ]$ is called extremal if $[x,[x,y]]$ is a multiple of $x$ for all $y$. In this paper we characterize the (finitary) symplectic Lie algebras as simple Lie algebras generated by their extremal elements satisying the condition that any two noncommuting extremal elements $x,y$ generate an $\mathfrak{sl}_2$ and any third extremal element $z$ commutes with at least one extremal element in this $\mathfrak{sl}_2$.

math.RA↗
arXiv2017

The geometry of hyperbolic lines in polar spaces

In this paper we consider partial linear spaces induced on the point set of a polar space, but with as lines the hyperbolic lines of this polar space. We give some geometric characterizations of these and related spaces. The results have applications in group theory, in the theory of Lie algebras and in graph theory.

math.CO↗
arXiv2014

Recovering the Lie algebra from its extremal geometry

An element $x$ of a Lie algebra $L$ over the field $F$ is extremal if $[x,[x,L]]=Fx$. Under minor assumptions, it is known that, for a simple Lie algebra $L$, the extremal geometry ${\cal{E}}(L)$ is a subspace of the projective geometry of $L$ and either has no lines or is the root shadow space of an irreducible spherical building $Δ$. We prove that if $Δ$ is of simply-laced type, then $L$ is a quotient of a Chevalley algebra of the same type.

math.RA↗