Actions of the additive group G a on certain noncommutative deformations of the plane

Ivan Kaygorodov; Samuel A. Lopes; Farukh Mashurov

Communications in Mathematics (2021)

  • Volume: 29, Issue: 2, page 269-279
  • ISSN: 1804-1388

Abstract

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We connect the theorems of Rentschler [rR68] and Dixmier [jD68] 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 ) 〉 , 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 A h . We also compute the Makar-Limanov invariant of absolute constants of A h over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of A h .

How to cite

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Kaygorodov, Ivan, Lopes, Samuel A., and Mashurov, Farukh. "Actions of the additive group $ {G}_a$ on certain noncommutative deformations of the plane." Communications in Mathematics 29.2 (2021): 269-279. <http://eudml.org/doc/297521>.

@article{Kaygorodov2021,
abstract = {We connect the theorems of Rentschler [rR68] and Dixmier [jD68] 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 \[A \_h=\langle x, y\mid yx-xy=h(x)\rangle \,,\] where $h$ is an arbitrary polynomial in $x$. In the second part of the paper we consider a field $\mathbb \{F\}$ of prime characteristic and study $\mathbb \{F\}[t]$comodule algebra structures on $A _h$. We also compute the Makar-Limanov invariant of absolute constants of $A _h$ over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of $A _h$.},
author = {Kaygorodov, Ivan, Lopes, Samuel A., Mashurov, Farukh},
journal = {Communications in Mathematics},
keywords = {Derivations; iterative higher derivations; rings of differential operators; Weyl algebra},
language = {eng},
number = {2},
pages = {269-279},
publisher = {University of Ostrava},
title = {Actions of the additive group $ \{G\}_a$ on certain noncommutative deformations of the plane},
url = {http://eudml.org/doc/297521},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Kaygorodov, Ivan
AU - Lopes, Samuel A.
AU - Mashurov, Farukh
TI - Actions of the additive group $ {G}_a$ on certain noncommutative deformations of the plane
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 2
SP - 269
EP - 279
AB - We connect the theorems of Rentschler [rR68] and Dixmier [jD68] 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 \[A _h=\langle x, y\mid yx-xy=h(x)\rangle \,,\] where $h$ is an arbitrary polynomial in $x$. In the second part of the paper we consider a field $\mathbb {F}$ of prime characteristic and study $\mathbb {F}[t]$comodule algebra structures on $A _h$. We also compute the Makar-Limanov invariant of absolute constants of $A _h$ over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of $A _h$.
LA - eng
KW - Derivations; iterative higher derivations; rings of differential operators; Weyl algebra
UR - http://eudml.org/doc/297521
ER -

References

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