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Picard-Vessiot extensions of real differential fields. (English) Zbl 1447.12004
Let \(K\) be an ordinary differential field of characteristic zero with a field of constants \(k\), \(P\) be a linear differential operator over \(K\) and \(\eta_{1},\ldots,\eta_{n}\) is a fundamental system of zeros of \(P\). The field \(L=K\langle\eta_{1},\ldots,\eta_{n}\rangle\) is called a Picard-Vessiot extension of the field \(K\), if the constants \(L\) and \(K\) coincide. When \(k\) is algebraically closed, according to the differential Galois theory, such an extension for \(P\) always exists and it is unique up to \(K\)-differential isomorphism. In this case, the operator \(P\) can be assigned a group G of \(K\)-differential automorphisms by setting \(G=\mathrm{DGal}(L/K)\) which has the structure of a linear algebraic group defined over \(k\). If the field \(k\) is not algebraically closed, then the situation is substantially complicated. For example, uniqueness may be violated. In the paper, the authors investigate a such case when \(K\) is formally real differential field with real closed field of constants \(k\). In this case for any given connected semi-simple linear algebraic group \(G\) defined over \(k\), there exists (see [T. Crespo et al., Math. Ann. 365, No. 1–2, 93–103 (2016; Zbl 1344.34096), 1, Proposition 3.3]) a linear differential operator \(P\) over \(K\) and a formally real Picard-Vessiot extension \(L/K\) for it such that \(G=\mathrm{DGal}(L/K)\). The paper discusses in detail the case when a complexification \(H(H:=G\times_{k}\bar{k})\) of \(G\) is one of the classical groups \(\mathrm{SL}\), \(\mathrm{SU}\), \(\mathrm{SO}\) or \(\mathrm{Sp}\). “The inspection of the different cases shows that there is no general pattern.”
MSC:
12H05 Differential algebra
13B05 Galois theory and commutative ring extensions
14P05 Real algebraic sets
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
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