The Jacobian algebras

Let Pn := K[x1, . . . , xn] be a polynomial algebra over a field K of characteristic zero. The Jacobian algebra An is the subalgebra of EndK(Pn) generated by the Weyl algebra An := D(Pn) = K〈x1, . . . , xn, ∂1, . . . , ∂n〉 and the elements (∂1x1) −1, . . . , (∂nxn) −1 ∈ EndK(Pn). The algebra An appears naturally in study of the group of automorphisms of Pn. The algebra An is large since it contains a ring M∞(K) := lim −→d of infinite dimensional matrices and all (formal) integrodifferential operators as all ∫ i = xi(∂ixi) −1 ∈ An. Surprisingly, the algebras An and An have little in common: the algebra An is neither left nor right Noetherian (even contains infinite direct sums of nonzero left and right ideals); not simple; not a domain; contains nilpotent elements; local; prime; central; self-dual; GK (An) = 3n; cl.K.dim(An) = n; has only finitely many, say sn, ideals (2−n+ ∑n i=1 2 (ni) ≤ sn ≤ 2 2n) which are found explicitly (they commute, IJ = JI; each of them is an idempotent ideal, I2 = I, and GK (An/I) = 3n if I 6= An). Spec(An) is found, it contains exactly 2 elements, each nonzero prime ideal is a unique sum of primes of height 1 (and any such a sum is a prime ideal). Each nonzero ideal is a unique product and a unique intersection of incomparable primes; moreover in both presentations the primes are the same, they are the minimal primes over the ideal. The group of units An of An is huge (An ⊇ K ∗ × ((Zn)(Z) ⋉GL∞(K))). An has only one faithful simple module, namely Pn.