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Trees of semi-simple algebras

Jan Adriaenssens·Lieven Le Bruyn
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Abstract

To a tree of semi-simple algebras we associate a qurve (or formally smooth algebra) S. We introduce a Zariski- and etale quiver describing the finite dimensional representations of S. In particular, we show that all quotient varieties of the etale quiver have a natural Poisson structure induced by a double Poisson algebra structure on a certain universal localization of its path algebra. Explicit calculations are included for the group algebras of the arithmetic groups (P)SL2(Z) and GL2(Z) but the methods apply as well to congruence subgroups.

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Jan Adriaenssens, Lieven Le Bruyn. 2005-07-25. Trees of semi-simple algebras. https://arxiv.org/abs/math/0507503

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