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Algebra and Discrete Mathematics, 2013, Volume 15, Issue 1, Pages 96–126
(Mi adm414)
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This article is cited in 5 scientific papers (total in 5 papers)
RESEARCH ARTICLE
Automorphic equivalence of the representations of Lie algebras
I. Shestakov, A. Tsurkov Institute of Mathematics and Statistics, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo - SP - Brasil - CEP 05508-090
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.
Keywords:
universal algebraic geometry, representations of Lie algebras, automorphic equivalence.
Received: 15.12.2012 Revised: 15.12.2012
Citation:
I. Shestakov, A. Tsurkov, “Automorphic equivalence of the representations of Lie algebras”, Algebra Discrete Math., 15:1 (2013), 96–126
Linking options:
https://www.mathnet.ru/eng/adm414 https://www.mathnet.ru/eng/adm/v15/i1/p96
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