Claim: The Dixmier conjecture is false in rank 3.

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Claim: The Dixmier conjecture is false in rank 3.

Let A_3 = C<x,y,z,Dx,Dy,Dz> be the third Weyl algebra. Define phi on x,y,z by

phi(x) = (1+xy)^3 z + y^2(1+xy)(4+3xy)

phi(y) = y + 3x(1+xy)^2 z + 3xy^2(4+3xy)

phi(z) = 2x - 3x^2y - x^3z.

Let F = (phi(x),phi(y),phi(z)), and define

phi(D_i) = sum_k (J_F^{-1})_{ki} D_k.

Here det(J_F) = -2, so J_F^{-1} has polynomial entries and phi is an endomorphism of A_3. It is injective because A_3 is simple.

It is not surjective: F is not injective, since

F(0,0,-1/4)
= F(1,-3/2,13/2)
= F(-1,3/2,13/2)
= (-1/4,0,0).

Moreover, the centralizer of phi(x), phi(y), phi(z) in A_3 is C[x,y,z]. Hence, if phi were an automorphism, then

C[x,y,z] = C[phi(x),phi(y),phi(z)],

which would make F a polynomial automorphism, contradicting the displayed non-injectivity.

Therefore phi is an injective, non-surjective endomorphism of A_3, giving a counterexample to the Dixmier conjecture in rank 3.