Borel subalgebras of Lie algebras of vector fields

TL;DR

This paper introduces and classifies integrable Borel subalgebras within the Lie algebra of automorphisms of affine varieties, focusing on toric affine surfaces and their automorphism groups.

Contribution

It defines integrable Borel subalgebras and provides a classification for toric affine surfaces, extending previous work on automorphism groups.

Findings

Identified integrable Borel subalgebras as tangent algebras of Borel subgroups

Classified these subalgebras for toric affine surfaces

Extended understanding of automorphism groups of affine varieties

Abstract

In [I. Arzhantsev and M. Zaidenberg, Borel subgroups of the automorphism groups of affine toric surfaces, arXiv:2507.09679 (2025)] we described the Borel subgroups and maximal solvable subgroups of the automorphism groups of affine toric surfaces. In the present paper, we introduce the notion of an integrable Borel subalgebra in the Lie algebra of the automorphism group of an affine variety. We show that they are precisely the tangent algebras of the Borel subgroups. We classify the integrable Borel subalgebras in the Lie algebras of the automorphism groups of toric affine surfaces, notably of the affine plane and its cyclic quotients.