Lie algebras of derivations and affine algebraic geometry over fields of characteristic \(0\)
The purpose of this paper is the study of derivation Lie algebras \(Der(A)\) of finitely generated commutative algebras \(A\) over an algebraically closed field of characteristic 0. The basic models are `Lie algebras of vector fields' \(Der(A_X)\) for the rings \(A_X\) of coordinates of affine varieties \(X\). The main result states that two normal affine varieties \(X\) and \(X'\) are isomorphic if and only if their Lie algebras \(Der(A_X)\) and \(Der (A_{X'})\) are so, which is an analog of the well-known theorem by \textit{M. E. Shanks} and \textit{L. E. Pursell} [Proc. Am. Math. Soc. 5, 468-472 (1954; Zbl 0055.42105)] in the case of smooth manifolds. The key idea is to relate certain Lie subalgebras, described by purely Lie theoretic properties, to geometric objects like points, subvarieties, or germs of subvarieties. As a further result it is deduced that an affine variety \(X\) is smooth if and only if the Lie algebra \(Der(A_X)\) is simple, which is an extension of a result by \textit{D. A. Jordan} [J. Lond. Math. Soc., II. Ser. 33, No. 2, 33-39 (1986; Zbl 0591.17007)].
scientific article; zbMATH DE number 931957
scientific article