On restriction of roots on affine $$T$$ T -varieties

Abstract

Let \(X\) be a normal affine algebraic variety with a regular action of a torus \(\mathbb {T}\) and \(T\subset \mathbb {T}\) be a subtorus. We prove that each root of \(X\) with respect to \(T\) can be obtained by restriction of some root of \(X\) with respect to \(\mathbb {T}\). This allows to give an elementary proof of the description of roots of the affine Cremona group. Several results on restriction of roots in the case of a subtorus action on an affine toric variety are obtained.