Automorphic equivalence of the representations of Lie algebras

• Main Authors: Shestakov, I., Tsurkov, A.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2013
• Series: Algebra and Discrete Mathematics
• Online Access: http://dspace.nbuv.gov.ua/handle/123456789/152257

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 (k) = 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.