Actions of the additive group G_a on certain noncommutative deformations of the plane

Abstract

We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y | y x − x y = h ( x ) 〉 , {A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle , , where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah . We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah .

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

“…The Jacobian (and hence also the Dixmier) conjecture has many other equivalent forms, one of which concerns locally nilpotent derivations on polynomial rings and on Poisson algebras (see Subsection 1.8). In fact, there seems to be a close connection in general between locally nilpotent derivations of (possibly noncommutative or nonassociative) algebras and automorphism groups, which has motivated our note in [81] connecting the theorems of Rentschler [115] and Dixmier [48] on locally nilpotent derivations and automorphisms of the polynomial ring F[x 1 , x 2 ] and of the Weyl algebra A 1 (F).…”

“…} is an abelian Lie algebra of locally nilpotent derivations of A h . This motivated our study in [81] of the locally nilpotent derivations of A h and more generally, in case char(F) > 0, of Hasse-Schmidt higher derivations of A h . In fact, in [81] we connected the theorems of Rentschler [115] and Dixmier [48] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1 by establishing the same type of results for the family of algebras A h .…”