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Automorphisms of the endomorphism semigroups of free linear algebras of homogeneous varieties

机译：同质变分自由线性代数的内同构半群的自同构

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We consider homogeneous varieties of linear algebras over an associative-commutative ring K with 1, i.e., the varieties in which free algebras are graded. Let F F(x(1),..., x(n)) be a free algebra of some variety Theta of linear algebras over K freely generated by a set X {x(1),..., x(n)), End F be the semigroup of endomorphisms of F, and Aut End F be the group of automorphisms of the semigroup End F. We investigate the structure of the group Aut End F and its relation to the algebraic and categorical equivalence of algebras from Theta. We define a wide class of R1MF-domains containing, in particular, Bezout domains, unique factorization domains, and some other domains. We show that every automorphism Phi of semigroup End F, where F is a free finitely generated Lie algebra over an R1MF-domain, is semi-inner. This solves the Problem 5.1 left open in [G. Mashevitzky, B. Plotkin, E. Plotkin, Automorphisms of the category of free Lie algebras, J. Algebra 282 (2004) 490-512]. As a corollary, semi-inner character of all automorphisms of the category of free Lie algebras over R1MF-domains is obtained. Relations between categorical and geometrical equivalence of Lie algebras over R1MF-domains are clarified. The group Aut End F for the variety of in-nilpotent associative algebras over R1MF-domains is described. As a consequence, a complete description of the group of automorphisms of the full matrix semigroup of n x n matrices over R1MF-domains is obtained. We give an example of the variety Theta of linear algebras over a Dedekind domain such that not all automorphisms of Aut End F are quasi-inner. The results obtained generalize the previous studies of various special cases of varieties of linear algebras over infinite fields. (C) 2008 Elsevier Inc. All rights reserved.
机译：我们考虑在具有1的缔合-交换环K上的线性代数的同质变体，即自由代数被分级的变体。令FF（x（1），...，x（n））是由一组X {x（1），...，x（n）自由生成的K上线性代数的各种Theta的自由代数），End F是F的内同构的半群，而Aut End F则是F的半群同构。 。我们定义了一大类R1MF域，尤其是包含Bezout域，唯一分解域和其他一些域的R1MF域。我们表明，半群End F的每个自同构Phi都是半内在的，其中F是在R1MF域上的自由有限生成的Lie代数。这解决了在[G. Mashevitzky，B.Plotkin，E.Plotkin，自由李代数类别的自同构，J.Algebra 282（2004）490-512]。作为推论，获得了R1MF域上的自由李代数类别的所有自同构的半内在特征。阐明了李代数在R1MF域上的范畴和几何等价关系。描述了R1MF域上各种幂等缔合代数的Aut End F组。结果，获得了在R1MF域上的n x n矩阵的完整矩阵半群的自同构群的完整描述。我们给出一个Dedekind域上线性代数的变种Theta的示例，使得并非Aut End F的所有自同构都是拟内的。获得的结果概括了以往对无限域上线性代数的各种特殊情况的研究。 （C）2008 Elsevier Inc.保留所有权利。

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