The group of automorphisms of the Jacobian algebra An

The Jacobian algebra A(n) is obtained from the Weyl algebra A(n) by inverting (not in the sense of Ore) of certain elements (A(n) is neither a Noetherian algebra nor a domain, A(n) contains the algebra K(x(1), ... , x(n), a/ax(1), ... , a/ax(n), integral(1) , ... , integral(n)) of polynomial integro-differential operators). The group of automorphisms G(n) of the Jacobian algebra A(n) is found (G(n) is a huge group): G(n) = S-n x (T-n x Xi(n)) x Inn(A(n)) superset of S-n x (T-n x (Z(n))((z)) x GL(infinity)(K) x ... x GL(infinity) (K)./2(n)-1 times G(1) similar or equal to (T-1 x Z((z))) x GL(infinity)(K), where S-n is the symmetric group, T-n is the n-dimensional algebraic torus, Xi(n) similar or equal to Z(n) is a group given explicitly, Inn(A(n)) is the group of inner automorphisms of A(n) (which is huge), GL(infinity)(K) is the group of invertible infinite dimensional matrices, and (Z(n))((z)) is a direct sum of Z copies of the free abelian group Z(n). This result may help in understanding of the structure of the groups of automorphisms of the Weyl algebra A(n) and the polynomial algebra P-2n. Explicit generators are found for the group G(1). The stabilizers in G(n) of all the ideals of A(n) are found, they are subgroups of finite index in G(n). It is shown that the group G(n) has trivial center. An explicit inversion formula is given for the elements of G(n). Defining relations are found for the algebra A(n). (C) 2011 Elsevier B.V. All rights reserved.