Observable subgroups of algebraic monoids

TL;DR

This paper extends the concept of observable subgroups from algebraic groups to algebraic monoids, characterizing their properties and providing conditions for observability based on invariant functions and closure.

Contribution

It introduces the notion of observable subgroups within algebraic monoids and establishes criteria involving invariant functions and closure properties, expanding the theory from groups to monoids.

Findings

H is observable in M iff H is closed in M and has enough H-semiinvariant functions.

A closed, normal subgroup H of G is observable iff it is closed in M, with a determinant character.

The quotient M/_{aff} H is an affine algebraic monoid scheme with unit group G/H.

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 $\chi: M \to K$ such that $H\subset ker(\chi)$. 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.