### 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.

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