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K1() and the group of automorphisms of the algebra of one-sided inverses of a polynomial algebra in two variables

Part of: Affine geometry Birational geometry Curves

Published online by Cambridge University Press:  24 May 2012

V.V. Bavula
V.V. Bavula*
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK, [email protected]
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Abstract

Explicit generators are found for the group G2 of automorphisms of the algebra of one-sided inverses of a polynomial algebra in two variables over a field. Moreover, it is proved that

where S2 is the symmetric group, is the 2-dimensional algebraic torus, E∞() is the subgroup of GL∞() generated by the elementary matrices. In the proof, we use and prove several results on the index of an operator. The final argument is the proof of the fact that K1() ≃ K*. The algebras and are noncommutative, non-Noetherian, and not domains.

Keywords

the group of automorphismsthe inner automorphismsthe Fredholm operatorsthe index of an operatorK1(S1)the semi-direct product of groupsthe minimal primes.

MSC classification

Secondary: 14E07: Birational automorphisms, Cremona group and generalizations 14H37: Automorphisms 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 14R15: Jacobian problem
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 3 , December 2012 , pp. 583 - 601
Copyright
Copyright © ISOPP 2012

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