Geometric constructions for loops

Stefano Pasotti
Universit\`a degli Studi di Brescia

Silvia Pianta
Universit\`a Cattolica del Sacro Cuore di Brescia

Elena Zizioli
Universit\`a degli Studi di Brescia



Content: Over the last ten years several authors established and studied the correspondence between geometric structures, loops and regular permutation sets. These relationships were first introduced in [2] in order to associate to the point-set of a general hyperbolic geometry over a Euclidean field the algebraic structure of a so-called \emph{Bruck loop} (or \emph{K-loop}) and use it for a ``coordinatization'' of the geometry. This research is placed in the line of investigations aiming at employing geometric structures and the related insight in order to build up loops and to study their algebraic properties. In particular we present here two techniques (see [4], [3]). The first one is a generalization of the idea of describing loops by means of \emph{complete graphs with a suitable edge-colouring} already considered in a special case in [1]. In a more general setting we employ complete directed graphs, we show how to relate a loop to suitable edge-colouring of these graphs and we find conditions characterizing graphs giving rise to the same loop, to isomorphic and to isotopic loops. The second one exploits techniques related to \emph{transversals and sections of groups}. Here we consider the group $\mathrm{PGL}_2(\mathbf{K})$ identified with the 3-dimensional projective space $\mathrm{PG}(3,\mathbf{K})$ deprived of the point-set of a ruled quadric $\mathcal{Q}$ and we consider a suitable subgroup $\mathcal{D}$. In this context we search for geometrically relevant subsets of $\mathrm{PG}(3,\mathbf{K})\setminus\mathcal{Q}$ which are a complete set of representatives for the left cosets of $\mathcal{D}$ and equip them with the structure of (left) loops. In particular we consider both the case $\mathbf{K}$ is an Euclidean field (characterizing loops arising from planes of the projective space and from the tangent semicone to $\mathcal{Q}$ from the point $1$) and the case $\mathbf{K}=\mathrm{GF}(q)$, $q$ odd (considering different types of subgroups and transversals which are geometrically well characterized). \vspace{5mm} \noindent{\bf References:} \begin{itemize} \item[{[1]}] Helmut Karzel, Silvia Pianta, and Elena Zizioli, \emph{{Loops, reflection structures and graphs with parallelism}}, Results Math. \textbf{42} (2002), no.~1-2, 74--80. \item[{[2]}] Helmut Karzel and Heinrich Wefelscheid, \emph{{Groups with an involutory antiautomorphism and {$K$}-loops; application to space-time-world and hyperbolic geometry. {I}}}, Results Math. \textbf{23} (1993), no.~3-4, 338--354. \item[{[3]}] Stefano Pasotti, Silvia Pianta, and Elena Zizioli, \emph{A geometric environment for building up loops}, Results Math. (2015), To appear. \item[{[4]}] Stefano Pasotti and Elena Zizioli, \emph{{Loops, regular permutation sets and colourings of directed graphs}}, J. Geom. \textbf{106} (2015), 35--45. \end{itemize}

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