Lie algebras of vector fields on smooth aﬃne varieties
We reprove the results of Jordan  and Siebert  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|
|Journal||Communications in Algebra|
Billig, Y, & Futorny, V. (Vyacheslav). (2018). Lie algebras of vector fields on smooth aﬃne varieties. Communications in Algebra, 1–17. doi:10.1080/00927872.2017.1412456