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The author has shown [Illinois
J. Math. (1969)] that a connected affine algebraic group over an algebraically closed
field of characteristic 0 has a universal affine covering if and only if its radical is
unipotent. The attempt to construct universal coverings of arbitrary connected affine
algebraic groups forces the acceptance of pro-affine algebraic groups, whose Hopf
algebras of polynomial functions are not necessarily finitely generated. This is a
motivation for extending the covering theory over the larger category of pro-affine
algebraic groups.
As we shall see here, the basic methods and results from the theory of affine
algebraic groups extend easily and smoothly so as to yield the appropriate results
concerning coverings of pro-affime algebraic groups over an algebraically
closed field of characteristic 0. In particular, the Lie algebras of these groups
can be used for obtaining a simple and natural construction of universal
coverings.
Our reconsideration of the covering theory also fills a gap in the theory for affine
groups. One of our main results gives a characterization, in algebraic-geometric
terms, of those ‘space coverings’ which arise from group coverings. The simplicity of
this characterization is undoubtedly due to the assmmption that the base field be of
cfiaracteristic 0.
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