The group of automorphisms of the Jacobian algebra AnAn

The Jacobian   algebra AnAn is obtained from the Weyl algebra AnAn by inverting (not in the sense of Ore) of certain elements (AnAn is neither a Noetherian algebra nor a domain, AnAn contains the algebra K〈x1,…,xn,∂∂x1,…,∂∂xn,∫1,…,∫n〉 of polynomial integro-differential operators). The group of automorphisms GnGn of the Jacobian algebra AnAn is found (GnGn is a huge group): Gn=Sn⋉(Tn×Ξn)⋉Inn(An)⊇Sn⋉(Tn×(Zn)(Z))⋉GL∞(K)⋉⋯⋉GL∞(K)︸2n−1times,G1≃(T1×Z(Z))⋉GL∞(K), where SnSn is the symmetric group, TnTn is the nn-dimensional algebraic torus, Ξn≃ZnΞn≃Zn is a group given explicitly, Inn(An) is the group of inner automorphisms of AnAn (which is huge), GL∞(K) is the group of invertible infinite dimensional matrices, and (Zn)(Z)(Zn)(Z) is a direct sum of ZZ copies of the free abelian group ZnZn. This result may help in understanding of the structure of the groups of automorphisms of the Weyl algebra AnAn and the polynomial algebra P2nP2n. Explicit generators are found for the group G1G1. The stabilizers in GnGn of all the ideals of AnAn are found, they are subgroups of finite   index in GnGn. It is shown that the group GnGn has trivial center. An explicit inversion formula is given for the elements of GnGn. Defining relations are found for the algebra AnAn.