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The Group of Automorphisms of the Algebra of one-Sided Inverses of a Polynomial Algebra. II

Published online by Cambridge University Press:  03 June 2015

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

The algebra of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra is isomorphic to the algebra

of scalar integro-differential operators provided that char(K) = 0. Ignoring the non-Noetherian property, the algebra belongs to a family of algebras like the nth Weyl algebra An and the polynomial algebra P2n. Explicit generators are found for the group Gn of automorphisms of the algebra and for the group of units of (both groups are huge). An analogue of the Jacobian homomorphism AutK-alg (Pn) → K* is introduced for the group Gn (notice that the algebra is non-commutative and neither left nor right Noetherian). The polynomial Jacobian homomorphism is unique. Its analogue is also unique for n > 2 but for n = 1, 2 there are exactly two of them. The proof is based on the following theorem that is proved in the paper:

Keywords

group of automorphismsgroup generatorsinner automorphismsFredholm operatorsindex of an operatorsemi-direct and exact products of groupsPrimary 16W2014E0714H3714R1014R15
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 543 - 580
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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