The Group of Automorphisms of the Algebra of one-Sided Inverses of a Polynomial Algebra. II. Issue 2 (3rd June 2015)

• Record Type: Journal Article
• Authors: Bavula, V. V.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 58:Issue 2(2015)
• Page Start: 543
• Page End: 580
• Publication Date: 2015-06-03
• Subjects: group of automorphisms, -- group generators, -- inner automorphisms, -- Fredholm operators, -- index of an operator, -- semi-direct and exact products of groups; Primary 16W20, -- 14E07, -- 14H37, -- 14R10, -- 14R15; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091514000303
• Languages: English
• ISSNs: 0013-0915

Abstract

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 n th 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 Aut K -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: