Connected linear groups as differential Galois groups.

*(English)*Zbl 0867.12004This paper gives an algebraic and essentially constructive proof of the following theorem: Let \(C\) be an algebraically closed field of characteristic zero, \(G\) a connected linear algebraic group over \(C\), and \(k\) a differential field with constant field \(C\) and finite (non-zero) transcendence degree over \(C\). Then \(G\) can be realized as the differential Galois group of a Picard-Vessiot extension of \(k\). As the authors explain in their careful historical introduction, this result is known by analytic methods for the case \(C=\mathbb{C}\) and \(k=\mathbb{C}(x)\), and that the program for the algebraic solution, and a number of key steps, are due to J. Kovacic.

While explaining their solution requires the concepts and notation of Kovacic’s theory, which can not be summarized here, the basic idea could be characterized as follows: an extension is to be constructed for the group \(G\). If it fails to be Picard-Vessiot with respect to a certain differential equation, this can be explained in terms of the presence of a certain element in the Lie algebra of \(G\). By designing the equation to avoid this possibility, the result is obtained. For the key case of \(G\) semisimple with Lie algebra \({\mathcal G}\), this is done by finding a faithful \({\mathcal G}\) module \(V\) and an appropriate pair of generators \((A_0,A_1)\) of \({\mathcal G}\subset gl(V)\) such that the equation \(Y'=(A_0+ A_1)Y\) has group \(G\).

While explaining their solution requires the concepts and notation of Kovacic’s theory, which can not be summarized here, the basic idea could be characterized as follows: an extension is to be constructed for the group \(G\). If it fails to be Picard-Vessiot with respect to a certain differential equation, this can be explained in terms of the presence of a certain element in the Lie algebra of \(G\). By designing the equation to avoid this possibility, the result is obtained. For the key case of \(G\) semisimple with Lie algebra \({\mathcal G}\), this is done by finding a faithful \({\mathcal G}\) module \(V\) and an appropriate pair of generators \((A_0,A_1)\) of \({\mathcal G}\subset gl(V)\) such that the equation \(Y'=(A_0+ A_1)Y\) has group \(G\).

Reviewer: A.R.Magid (Norman)