Lie algebras of vector fields on smooth affine varieties

ABSTRACT

We reprove the results of Jordan [18] and Siebert [30] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.

KEYWORDS:

• Lie algebra of vector fields
• smooth algebraic variety

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17B20
• 17B66
• Secondary: 13N15

Acknowledgments

Y. B. gratefully acknowledges the hospitality and excellent working conditions at the University of São Paulo where this work was done.