کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4603479 | 1336962 | 2008 | 25 صفحه PDF | دانلود رایگان |

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(x1,…,xn) be a free algebra of some variety Θ of linear algebras over K freely generated by a set X={x1,…,xn}, 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 Θ.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 Φ 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 m-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×n matrices over R1MF-domains is obtained.We give an example of the variety Θ 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.
Journal: Linear Algebra and its Applications - Volume 429, Issue 1, 1 July 2008, Pages 156-180