کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4603479 1336962 2008 25 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Automorphisms of the endomorphism semigroups of free linear algebras of homogeneous varieties
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
Automorphisms of the endomorphism semigroups of free linear algebras of homogeneous varieties
چکیده انگلیسی

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.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Linear Algebra and its Applications - Volume 429, Issue 1, 1 July 2008, Pages 156-180