Equivariant vector fields on non-trivial \(\text{SO}_n\)-torsors and differential Galois theory. (English) Zbl 1173.12001
Let \(F\) be a differential field with algebraically closed characteristic zero field of constants \(C\), and let \(E\) be a Picard–Vessiot (differential Galois) extension of \(F\). Then the group \(G\) of differential automorphisms of \(E\) over \(F\) has the structure of a linear algebraic group over \(C\). Let \(G_F = G \times_C F\); then \(E\) is the function field of an \(F\)-variety \(V\) which is a torsor for \(G_F\). In many constructed examples of Picard–Vessiot extensions in the literature, the torsor \(V\) is actually the trivial torsor \(G_F\). In this paper, the authors produce examples, for the case \(G=SO_n\), any \(n \geq 3\), where the torsor is non–trivial. The base field \(F\) (which depends on \(n\)) in the examples is generated over \(C\) by differential indeterminates.
Reviewer: Andy R. Magid (Norman)
MSC:
| 12H05 | Differential algebra |
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