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Dessins D'enfants, Surface Algebras, and Dessin Orders. (arXiv:1810.06750v1 [math.NT])
来源于:arXiv
We present a construction of an infinite dimensional associative algebra
which we call a \emph{surface algebra} associated in a unique way to a dessin
d'enfant. Once we have constructed the surface algebras we construct what we
call the associated \emph{dessin order}, which can be constructed in such a way
that it is the completion of the path algebra of a quiver with relations. We
then prove that the center and (noncommutative) normalization of the dessin
orders are invariant under the action of the absolute Galois group
$\mathcal{G}(\overline{\mathbb{Q}}/\mathbb{Q})$. We then describe the
projective resolutions of the simple modules over the dessin order and show
that one can completely recover the dessin with the projective resolutions of
the simple modules. Finally, as a corollary we are able to say that classifying
dessins in an orbit of $\mathcal{G}(\overline{\mathbb{Q}}/\mathbb{Q})$ is
equivalent to classifying dessin orders with a given normalization. 查看全文>>