Borel subalgebras of Lie algebras of vector fields

2025/10/20 by Ivan Arzhantsev, Mikhail Zaidenberg, Arzhantsev, Ivan +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2510.17223

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.

Citations

• Borel subgroups of the automorphism groups of affine toric surfaces
• Group Theoretical Characterizations of Rationality
• Automorphism groups of affine varieties and their Lie algebras
• Structure of connected nested automorphism groups
• Automorphisms of Veronese subalgebras of polynomial algebras and free Poisson algebras
• Automorphisms of affine Veronese surfaces
• Families of commuting automorphisms, and a characterization of the affine space
• On the geometry of the automorphism groups of affine varieties
• Lie algebras of vector fields on smooth affine varieties
• On the maximality of the triangular subgroup
• Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras
• Dixmier Groups and Borel Subgroups
• The groups of automorphisms of the Lie algebras of polynomial vector fields with zero or constant divergence
• On nilpotent and solvable Lie algebras of derivations

Cited by

• Regular Borel subalgebras of the Lie Algebra of Aut(\mathbbA²)

Related

• Borel subgroups of the automorphism groups of affine toric surfaces
• Automorphism groups of affine varieties and their Lie algebras
• Characterization of affine surfaces with a torus action by their automorphism groups
• Automorphism groups of non-normal affine toric surfaces
• On Lie's classification of subalgebras of vector fields on the plane