Finite-dimensional 2-generated Lie algebras of derivations on T-varieties

D. A. Matveev<sup>ab

a</sup> Faculty of Computer Science, HSE University, Moscow, Russia
^b Department of Higher Algebra, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

Abstract: We consider an affine algebraic variety with a torus action of complexity one. It is known that in this case homogeneous locally nilpotent derivations on the algebra of functions of this variety are defined in terms of a polyhedral divisor. In the present paper, a formula is obtained for multiple commutators of two homogeneous locally nilpotent derivations with at most one derivation of horizontal type. Using the obtained formula, a criterion is derived for finite dimensionality of Lie algebras generated by a pair of homogeneous locally nilpotent derivations in the Lie algebra of all derivations of the algebra of functions on a variety with a torus action.

Keywords: T-variety, graded algebra, locally nilpotent derivation, Lie algebra, commutator of derivations.

UDC: 512.554.35

MSC: 35R30

Received: 14.01.2024
Revised: 02.02.2025
Accepted: 25.02.2025

DOI: 10.33048/smzh.2025.66.311

 English version: Siberian Mathematical Journal, 2025, 66:3, 715–727