K1(S1) and the group of automorphisms of the algebra S2 of one-sided inverses of a polynomial algebra in two variables

Explicit generators are found for the group G2 of automorphisms of the algebra S2 of one-sided inverses of a polynomial algebra in two variables over a field of characteristic zero. Moreover, it is proved that G2 ≃ S2 ⋉ T 2 ⋉ Z ⋉ ((K∗ ⋉ E∞(S1))⊠GL∞(K) (K ∗ ⋉ E∞(S1))) where S2 is the symmetric group, T is the 2-dimensional torus, E∞(S1) is the subgroup of GL∞(S1) generated by the elementary matrices. In the proof, we use and prove several results on the index of operators, and the final argument in the proof is the fact that K1(S1) ≃ K∗ proved in the paper. The algebras S1 and S2 are noncommutative, non-Noetherian, and not domains. The group of units of the algebra S2 is found (it is huge).