Automorphisms of the Weyl Algebra

What is it about?

The authors present and discuss their conjecture that the automorphism group of the $n$-complex Weyl algebra, $A_{n,\Bbb C}$, is isomorphic to the automorphism group of the corresponding Poisson algebra $P_{n,\Bbb C}$, the latter algebra being the polynomial ring ${\Bbb C}[x_1,\dots,x_{2n}]$ equipped with the Poisson bracket such that $\{x_i,x_j\}=\delta_{i,n+j}-\delta_{i+n,j}$ for all $i, j$. This conjecture has a positive answer when $n=1$, because the groups $\roman{Aut}(A_{1,\Bbb C})$ and $\roman{Aut}(P_{1,\Bbb C})$ are known explicitly through work of J. Dixmier [Bull. Soc. Math. France 96 (1968), 209–242; MR0242897], L. Makar-Limanov [Bull. Soc. Math. France 112 (1984), no. 3, 359–363; MR0794737], and H. W. E. Jung [J. Reine Angew. Math. 184 (1942), 161–174; MR0008915]. A major argument in favor of the conjecture is the result that the subgroups of so-called tame automorphisms in $\roman{Aut}(A_{n,\Bbb C})$ and $\roman{Aut}(P_{n,\Bbb C})$ are naturally isomorphic, which the authors prove by reduction to positive characteristic. This approach also allows them to propose a hypothetical specific candidate for the conjectured isomorphism. Several variants and strengthenings of the main conjecture are discussed as well. Reviewed by K. R. Goodearl

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