The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra, II

The algebra Sn 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. Ignoring non-Noetherian property, the algebra Sn 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 Sn and for the group S∗n of units of Sn (both groups are huge). An analog of the Jacobian homomorphism Pn := AutK−alg(Pn) → K ∗ is introduced for the group Gn (notice that the algebra Sn is noncommutative and neither left nor right Noetherian). The polynomial Jacobian homomorphism is unique since Pn/[Pn,Pn] ≃ K ∗. 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: