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We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We then give a combinatorial classification of reductive monoids by means of the theory of spherical varieties.
In this short note we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.
Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ k[X] G such that f | Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if (1) there is a… (More)
We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley's structure theorem for algebraic groups.
Let M be an irreducible normal algebraic monoid with unit group G.
Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G = GantGaff ∼= (Gant ×Gaff )/Gaff ∩Gant, where Gant is the maximal anti-affine subgroup of G, and Gaff the maximal normal connected affine subgroup of G. In this paper we show that this decomposition extends to a decomposition M =… (More)
In this paper we generalize some constructions and results due to Cayley and Hilbert. We define the concept of Ω–process for an arbitrary algebraic monoid with zero and unit group G. Then we show how to produce from the process and for a linear rational representation of G, a number of elements of the ring of G-invariants, that is large enough as to… (More)
This paper is intended to very briefly introduce some computation problems in linear algebraic monoids. The theory of linear algebraic monoids was initiated independently in 1980 Ontario). This theory is a natural blend of algebraic groups, torus embeddings, and semigroups. It is a very active and fruitful research area in mathematics. Many other… (More)
Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T-module P(V). We determine when a projective toric subvariety X ⊂ P(V) is self-dual, in terms of the configuration of weights of V .
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… (More)