Automorphisms of the endomorphism semigroup of a polynomial algebra

作者：Belov Kanel A; Lipyanski R^*

来源：Journal of Algebra, 2011, 333(1): 40-54.

DOI：10.1016/j.jalgebra.2011.01.020

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

We describe the automorphism group of the endomorphism semi-group End(K[x(1,) ..., x(n)]) of ring K[x(1) ,..., x(n)] 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 phi is an element of Aut End(K[x(1), ... , x(n)]) then there exists a semi-linear automorphism s : K[x(1), ... , x(n)] -> K[x1, ... , x(n)] such that phi(g) = sogo s(-1) for any g is an element of End(K[x(1) , ... , xn]). This extends the result obtained by A. Berzins for an infinite field K.

• 出版日期2011-5-1