The group of order preserving automorphisms of the ring of differential operators on a Laurent polynomial algebra in prime characteristic

Abstract

Let K K be a field of characteristic p > 0 p>0 . It is proved that the group A u t o r d ( D ( L n ) ) \mathrm {Aut}_{ord}(\mathcal {D}(L_n)) of order preserving automorphisms of the ring D ( L n ) \mathcal {D}(L_n) of differential operators on a Laurent polynomial algebra L n := K [ x 1 ± 1 , … , x n ± 1 ] L_n:= K[x_1^{\pm 1}, \ldots , x_n^{\pm 1}] is isomorphic to a skew direct product of groups Z p n ⋊ A u t K ( L n ) {\mathbb {Z}}_p^n \rtimes \mathrm {Aut}_K(L_n) , where Z p {\mathbb {Z}}_p is the ring of p p -adic integers. Moreover, the group A u t o r d ( D ( L n ) ) \mathrm {Aut}_{ord}(\mathcal {D}(L_n)) is found explicitly. Similarly, A u t o r d ( D ( P n ) ) ≃ A u t K ( P n ) \mathrm {Aut}_{ord}(\mathcal {D}(P_n))\simeq \mathrm {Aut}_K(P_n) , where P n := K [ x 1 , … , x n ] P_n: =K[x_1, \ldots , x_n] is a polynomial algebra.