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

The algebra Sn in the title is obtained from a polynomial algebra Pn in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of Pn. Ignoring non-Noetherian property, the algebra Sn belongs to a family of algebras like the Weyl algebra An and the polynomial algebra P2n. The group of automorphisms Gn of the algebra Sn is found: Gn = Sn ⋉ T n ⋉ Inn(Sn) ⊇ Sn ⋉ T n ⋉ GL∞(K)⋉ · · · ⋉ GL∞(K) | {z } 2n−1 times =: G′n where Sn is the symmetric group, T is the n-dimensional torus, Inn(Sn) is the group of inner automorphisms of Sn (which is huge), and GL∞(K) is the group of invertible infinite dimensional matrices. This result may help in understanding of the structure of the groups of automorphisms of the Weyl algebra An and the polynomial algebra P2n. An analog of the Jacobian homomorphism: AutK−alg(P2n) → K ∗, the so-called global determinant is introduced for the group G′n (notice that the algebra Sn is noncommutative and neither left nor right Noetherian).