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arXiv · 0902.2207

Observable subgroups of algebraic monoids

Lex Renner·Alvaro Rittatore
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Abstract

A closed subgroup H of the affine, algebraic group G is called observable if G/H is a quasi-affine algebraic variety. In this paper we define the notion of an observable subgroup of the affine, algebraic monoid M. We prove that a subgroup H of G is observable in M if and only if H is closed in M and there are "enough" H-semiinvariant functions in K[M]. We show also that a closed, normal subgroup H of G (the unit group of M) is observable in M if and only if it is closed in M. In such a case there exists a determinant $χ: M \to K$ such that $H\subset ker(χ)$. As an application, we show that in this case the affinized quotient $M/_{aff} H$ of M by H is an affine algebraic monoid scheme with unit group G/H.

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Lex Renner, Alvaro Rittatore. 2009-02-12. Observable subgroups of algebraic monoids. https://arxiv.org/abs/0902.2207

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