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Borel subalgebras of Lie algebras of vector fields

Ivan Arzhantsev, Mikhail Zaidenberg

TL;DR

The paper develops a framework for integrable Borel subalgebras inside Lie algebras of automorphism ind-groups Aut(X), showing such subalgebras are precisely the tangent algebras to Borel subgroups and establishing a bijective Ad-conjugacy correspondence between integrable Borel subalgebras and Borel subgroups. It provides a complete classification in the toric affine surface setting, including the affine plane and its cyclic quotients X_{d,e}, and analyzes how these subgroups and their Lie algebras sit inside the ambient derivation algebras, highlighting contrasts with the finite-dimensional case. For small dimensions, it characterizes Borel subalgebras of Der(k[x]) and Lie(Aut(A^2)), including the existence of non-conjugate Borel subalgebras and the bigraded structure with Demazure roots; it also shows that in dimension three the triangular subalgebra is not maximal among solvable subalgebras, yielding integrable but non-Borel examples. The results illuminate the structure of automorphism groups of toric and affine varieties, providing tools to understand their solvable subgroups and corresponding tangent algebras, with implications for the geometry of toric singularities and their symmetries.

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.

Borel subalgebras of Lie algebras of vector fields

TL;DR

The paper develops a framework for integrable Borel subalgebras inside Lie algebras of automorphism ind-groups Aut(X), showing such subalgebras are precisely the tangent algebras to Borel subgroups and establishing a bijective Ad-conjugacy correspondence between integrable Borel subalgebras and Borel subgroups. It provides a complete classification in the toric affine surface setting, including the affine plane and its cyclic quotients X_{d,e}, and analyzes how these subgroups and their Lie algebras sit inside the ambient derivation algebras, highlighting contrasts with the finite-dimensional case. For small dimensions, it characterizes Borel subalgebras of Der(k[x]) and Lie(Aut(A^2)), including the existence of non-conjugate Borel subalgebras and the bigraded structure with Demazure roots; it also shows that in dimension three the triangular subalgebra is not maximal among solvable subalgebras, yielding integrable but non-Borel examples. The results illuminate the structure of automorphism groups of toric and affine varieties, providing tools to understand their solvable subgroups and corresponding tangent algebras, with implications for the geometry of toric singularities and their symmetries.

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.
Paper Structure (17 sections, 40 theorems, 100 equations)

This paper contains 17 sections, 40 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tangent Lie algebras of derivations
  4. Correspondence "ind-subgroups $\leftrightsquigarrow$ ind-subalgebras"
  5. Correspondence "Borel subgroups $\leftrightsquigarrow$ Borel subalgebras"
  6. Borel subalgebras of $\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb A}^n))$ for small $n$
  7. Borel subgroups of $\mathop{\rm Aut}({\mathbb A}^n)$
  8. The case of the affine line
  9. The case of the affine plane
  10. Borel subgroups
  11. $\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb A}^2))$ as a bigraded Lie algebra
  12. Borel subalgebras of $\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb A}^2))$
  13. The case of ${\mathbb A}^3$
  14. Subalgebras of $\mathop{\rm Lie}(\mathop{\rm Aut}(X_{d,e}))$
  15. Generalities
  16. ...and 2 more sections

Key Result

Theorem 1.1

The Lie algebra of triangular derivations of ${\Bbbk}[x,y]$ is maximal among the solvable subalgebras of $\mathop{\rm Lie}(\mathop{\rm Aut}({\mathbb A}^2))$. By contrast, the Lie algebra of triangular derivations of ${\Bbbk}[x,y,z]$ is not maximal among the solvable subalgebras of $\mathop{\rm Lie}(

Theorems & Definitions (80)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1: cf. KZ24
  • Theorem 2.2: MP14
  • Corollary 2.3
  • proof
  • Remark 2.4
  • ...and 70 more