Representations of the Lie algebra of vector fields on a sphere. (English) Zbl 1472.17024
Summary: For an affine algebraic variety \(X\) we study a category of modules that admit compatible actions of both the algebra \(A\) of functions on \(X\) and the Lie algebra of vector fields on \(X\). In particular, for the case when \(X\) is the sphere \(\mathbb{S}^2\), we construct a set of simple modules that are finitely generated over \(A\). In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational \(\mathrm{GL}_2\)-modules.
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B66 | Lie algebras of vector fields and related (super) algebras |
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