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<p>We consider an affine algebraic varietywith a torus action of complexity one.It is known that in this case homogeneous locally nilpotentderivations on the algebra of functions of this variety are definedin terms of a polyhedral divisor.In the present paper, a formula is obtained for multiplecommutators of two homogeneous locally nilpotentderivations with at most one derivationof horizontal type. Using the obtained formula,a&#xa0;criterion is derived for finite dimensionalityof Lie algebrasgenerated by a pair of homogeneous locally nilpotentderivations in the Lie algebra of all derivationsof the algebra of functions on a variety with a torus action.</p>

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Finite-Dimensional 2-Generated Lie Algebras of Derivations on T-Varieties

  • 期刊论文

  • 发表日期：

    2025年5月1日

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  • D. A. Matveev

摘要

We consider an affine algebraic varietywith a torus action of complexity one.It is known that in this case homogeneous locally nilpotentderivations on the algebra of functions of this variety are definedin terms of a polyhedral divisor.In the present paper, a formula is obtained for multiplecommutators of two homogeneous locally nilpotentderivations with at most one derivationof horizontal type. Using the obtained formula,a criterion is derived for finite dimensionalityof Lie algebrasgenerated by a pair of homogeneous locally nilpotentderivations in the Lie algebra of all derivationsof the algebra of functions on a variety with a torus action.