Automorphic equivalence of the representations of Lie algebras

Abstract

In this paper we research the algebraic geometry of the representations of Lie algebras over fixed field \(k\). We assume that this field is infinite and char \(\left(k\right) =0.\) We consider the representations of Lie algebras as \(2\)-sorted universal algebras. The representations of groups were considered by similar approach: as \(2\)-sorted universal algebras - in [3] and [2]. The basic notions of the algebraic geometry of representations of Lie algebras we define similar to the basic notions of the algebraic geometry of representations of groups (see [2]). We prove that if a field \(k\) has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. This result is similar to the result of [4], which was achieved for representations of groups. But we achieve our result by another method: by consideration of \(1\)-sorted objects. We suppose that our method can be more perspective in the further researches.