Automorphisms of the endomorphism semigroup of a polynomial algebra

A. Belov-Kanel; R. Lipyanski

doi:10.1016/j.jalgebra.2011.01.020

Automorphisms of the endomorphism semigroup of a polynomial algebra

A. Belov-Kanel, R. Lipyanski

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We describe the automorphism group of the endomorphism semigroup End(K[x1,...,xn]) of ring K[x1,...,xn] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative-associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15].More precisely, we prove that if. AutEnd(K[x1,...,xn]) then there exists a semi-linear automorphism s:K[x1,...,xn]K[x1,...,xn] such that (g)=sgs1 for any gEnd(K[x1,...,xn]). This extends the result obtained by A. Berzins for an infinite field K.

Original language	English
Pages (from-to)	40-54
Number of pages	15
Journal	Journal of Algebra
Volume	333
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2011.01.020
State	Published - 1 May 2011

Bibliographical note

Funding Information:
The authors are grateful to B. Plotkin for attracting their attention to this problem and interest to this work. The first author was supported by the Israel Science Foundation (grant No. 1178/06).

Keywords

Kronecker endomorphism
Polynomial algebra
Rank endomorphism
Semi-inner automorphism
Variety of commutative-associative algebras

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10.1016/j.jalgebra.2011.01.020

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title = "Automorphisms of the endomorphism semigroup of a polynomial algebra",

abstract = "We describe the automorphism group of the endomorphism semigroup End(K[x1,...,xn]) of ring K[x1,...,xn] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative-associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15].More precisely, we prove that if. AutEnd(K[x1,...,xn]) then there exists a semi-linear automorphism s:K[x1,...,xn]K[x1,...,xn] such that (g)=sgs1 for any gEnd(K[x1,...,xn]). This extends the result obtained by A. Berzins for an infinite field K.",

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author = "A. Belov-Kanel and R. Lipyanski",

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T1 - Automorphisms of the endomorphism semigroup of a polynomial algebra

AU - Belov-Kanel, A.

AU - Lipyanski, R.

N1 - Funding Information: The authors are grateful to B. Plotkin for attracting their attention to this problem and interest to this work. The first author was supported by the Israel Science Foundation (grant No. 1178/06).

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N2 - We describe the automorphism group of the endomorphism semigroup End(K[x1,...,xn]) of ring K[x1,...,xn] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative-associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15].More precisely, we prove that if. AutEnd(K[x1,...,xn]) then there exists a semi-linear automorphism s:K[x1,...,xn]K[x1,...,xn] such that (g)=sgs1 for any gEnd(K[x1,...,xn]). This extends the result obtained by A. Berzins for an infinite field K.

AB - We describe the automorphism group of the endomorphism semigroup End(K[x1,...,xn]) of ring K[x1,...,xn] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative-associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15].More precisely, we prove that if. AutEnd(K[x1,...,xn]) then there exists a semi-linear automorphism s:K[x1,...,xn]K[x1,...,xn] such that (g)=sgs1 for any gEnd(K[x1,...,xn]). This extends the result obtained by A. Berzins for an infinite field K.

KW - Kronecker endomorphism

KW - Polynomial algebra

KW - Rank endomorphism

KW - Semi-inner automorphism

KW - Variety of commutative-associative algebras

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Automorphisms of the endomorphism semigroup of a polynomial algebra

Abstract

Bibliographical note

Keywords

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