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Isotropy subgroups of homogeneous locally nilpotent derivations

Dmitriy Chunaev, Polina Evdokimova

Abstract

We say that a locally nilpotent derivations $δ$ is maximal if there are no inequivalent locally nilpotent derivations that commute with $δ$. The paper gives a description of isotropy groups of maximal homogeneous locally nilpotent derivations on affine toric varieties and on certain trinomial hypersurfaces. Moreover, the criteria for homogeneous locally nilpotent derivations to be maximal were obtained for these classes of varieties.

Isotropy subgroups of homogeneous locally nilpotent derivations

Abstract

We say that a locally nilpotent derivations is maximal if there are no inequivalent locally nilpotent derivations that commute with . The paper gives a description of isotropy groups of maximal homogeneous locally nilpotent derivations on affine toric varieties and on certain trinomial hypersurfaces. Moreover, the criteria for homogeneous locally nilpotent derivations to be maximal were obtained for these classes of varieties.

Paper Structure

This paper contains 10 sections, 23 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Locally nilpotent derivations
  3. Ind-groups
  4. Isotropy subgroups
  5. Toric varieties
  6. Commuting homogeneous LNDs on toric varieties
  7. Isotropy subgroups of homogeneous LNDs on toric varieties
  8. Trinomial hypersurfaces
  9. Commuting homogeneous LNDs on trinomial hypersurfaces
  10. Isotropy subgroups of homogeneous LNDs on trinomial hypersurfaces

Key Result

Proposition 2.5

F Let $r \in B$ be an arbitrary local slice then $B_{\delta(r)} = \nonumber \\ = A_{\delta(r)} [r]$. In particular, $\mathrm{tr.deg.}A = \mathrm{tr.deg.}B - 1$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Proposition 2.9
  • proof
  • ...and 54 more