The group of automorphisms of the Jacobian algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">A</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>
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 , ∂ ∂ x 1 , … , ∂ ∂ x n , ∫ 1 , … , ∫ 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 ⋉ ( T n × Ξ n ) ⋉ Inn ( A n ) ⊇ S n ⋉ ( T n × ( Z n ) ( Z ) ) ⋉ GL ∞ ( K ) ⋉ ⋯ ⋉ GL ∞ ( K ) ︸ 2 n − 1 times , G 1 ≃ ( T 1 × Z ( Z ) ) ⋉ GL ∞ ( K ) , where S n is the symmetric group, T n is the n -dimensional algebraic torus, Ξ n ≃ Z n is a group given explicitly, Inn ( A n ) is the group of inner automorphisms of A n (which is huge), GL ∞ ( 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 2 n . 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 .