Equivariant Embeddings of Algebraic Groups

Proposition 3 ([28], Proposition I.1). The correspondence S ↦→ Spec k[S] defines a bijection
between the set of finitely generated semi-groups S ⊂ X ∗ (T ) which generate X ∗ (T ) as a group and
the set of isomorphism classes of equivariant affine embeddings of T . Moreover, the morphisms
of equivariant affine embeddings correspond (in a contravariant way) to the inclusions between
semi-groups contained in X ∗ (T ).
We say a semi-group S ⊂ X ∗ (T ) is saturated if χ n ∈ S for some positive n ≥ 1 and χ ∈ X ∗ (T )
implies that χ ∈ S. This saturated condition corresponds to the normality of the associated torus
embedding, so that
Theorem 2 ([28], Theorem I.1). The correspondence S ↦→ Spec k[S] defines a bijection between
the set of finitely generated semi-groups S ⊂ X ∗ (T ) which generate X ∗ (T ) as a group and are
saturated and the set of equivariant normal affine embeddings of T .
Hence, affine toric varieties correspond to saturated semi-groups S of X ∗ (T ) which generate
X ∗ (T ) as a group. Now we classify all such saturated semi-groups of X ∗ (T ).
Regard X ∗ (T ) as a Z-lattice in the real vector space X ∗ (T ) R = X ∗ (T ) ⊗ Z R.
Definition 3. Let σ ⊂ X ∗ (T ) R . We call σ a convex rational polyhedral cone if σ = { ∑ N
i=1 λ ix i :
λ i ∈ R ≥0 , for all i} for some finite collection of elements x i ∈ X ∗ (T ).
We say σ is a strongly
convex rational polyhedral cone if it is a convex rational polyhedral cone and it does not contain
any non-zero linear subspace of X ∗ (T ) R .
We associate to a strongly convex rational polyhedral cone σ its dual cone, σ ∨ ⊂ X ∗ (T ) R , which
is defined as σ ∨ = {u ∈ X ∗ (T ) R : 〈u, v〉 ≥ 0 for all v ∈ σ}. Then σ ∨ is a convex, rational, polyhedral
cone in X ∗ (T ) R and (σ ∨ ) ∨ = σ. The correspondence σ ↦→ σ ∨ ∩ X ∗ (T ) defines a bijection between
the set of strongly convex rational polyhedral cones and the set of finitely generated semi-groups
in X ∗ (T ) which are saturated and generate X ∗ (T ) as a group by
Lemma 1 ([28], Gordan’s Lemma). Given a finite set of homogeneous linear integral inequalities,
the semi-group of integral solutions is finitely generated.
Therefore, we obtain the classification of affine toric varieties below.
13