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Locally finite solvable Lie algebras of derivations

Mikhail Zaidenberg

Abstract

Let X be an affine variety and L be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [arXiv:2507.09679] wondered whether L is itself locally finite. Here we present some criteria for the local finiteness of L. We also answer this question in the affirmative in the particular case where X is the affine plane.

Locally finite solvable Lie algebras of derivations

Abstract

Let X be an affine variety and L be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [arXiv:2507.09679] wondered whether L is itself locally finite. Here we present some criteria for the local finiteness of L. We also answer this question in the affirmative in the particular case where X is the affine plane.

Paper Structure

This paper contains 12 sections, 25 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. General affine varieties
  3. First reduction
  4. Second reduction.
  5. Third reduction
  6. Further reductions
  7. Case of the affine plane
  8. Triangular subalgebras of $\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb A}^2))$
  9. $\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb A}^2))$ as a bigraded Lie algebra
  10. Semisimple derivations and spectral decomposition
  11. Case of a non-algebraic semisimple derivation
  12. Case where all semisimple derivations of ${\mathfrak h}$ are algebraic

Key Result

Proposition 1.3

Let ${\mathfrak h}\subset\mathop{\rm Lie}(\mathop{\rm Aut}(X))$ be a Lie subalgebra as in Question ques-1. Then ${\mathfrak h}$ is finite-dimensional if and only if the derived ideal $[{\mathfrak h},{\mathfrak h}]$ is nilpotent. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (50)

  • Proposition 1.3: see Proposition \ref{['Prop 3']}
  • Theorem 1.4
  • proof
  • Remark 1.5
  • Theorem 1.6: see Theorem \ref{['mthm-A2']}
  • Lemma 2.2
  • Proposition 2.4: KZ24
  • Corollary 2.5
  • proof
  • Remark 2.6
  • ...and 40 more