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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Automorphisms of a free associative algebra of rank $ 2$. I


Author: Anastasia J. Czerniakiewicz
Journal: Trans. Amer. Math. Soc. 160 (1971), 393-401
MSC: Primary 16.60
DOI: https://doi.org/10.1090/S0002-9947-1971-0280549-1
MathSciNet review: 0280549
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Abstract: Let $ {R_{\left\langle 2 \right\rangle }} = R\left\langle {x,y} \right\rangle $ be the free associative algebra of rank 2, on the free generators $ x$ and $ y$, over $ R$ ($ R$ a field, a Euclidean domain, etc.). We prove that if $ \varphi $ is an automorphism of $ {R_{\left\langle 2 \right\rangle }}$ that keeps $ (xy - yx)$ fixed (up to multiplication by an element of $ R$), then $ \varphi $ is tame, i.e. it is a product of elementary automorphisms of $ {R_{\left\langle 2 \right\rangle }}$. This follows from a more general result about endomorphisms of $ {R_{\left\langle 2 \right\rangle }}$ via a theorem due to H. Jung [6] concerning automorphisms of a commutative and associative algebra of rank 2.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0280549-1
Keywords: Free associative algebra, endomorphisms, automorphisms, elementary automorphisms, tame automorphisms, wild automorphisms, polynomial rings
Article copyright: © Copyright 1971 American Mathematical Society