Observable subgroups of algebraic monoids

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 (the unit group of M) 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→k such that H⊆ker(χ). As an application, we show that in this case the affinized quotient of M by H is an affine algebraic monoid with unit group G/H.