Derivation algebras of toric varieties

Abstract

Normal affine algebraic varieties in characteristic 0 are uniquely determined (up to isomorphism) by the Lie algebra of derivations of their coordinate ring. This is not true without the hypothesis of normality. But, we show that (in general, non-normal) toric varieties defined by simplicial affine semigroups are uniquely determined by their Lie algebra if they are supposed to be Cohen-Macaulay of dimension at least 2 or Gorenstein of dimension 1. Moreover, every automorphism of the Lie algebra is induced from a unique automorphism of the variety and every derivation of the Lie algebra is inner, i.e., the first cohomology of the Lie algebra with coefficients in the adjoint representation vanishes.