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On adjoint Chevalley groups associated with completed Euclidean Lie algebras. (English) Zbl 0537.20009

Let \({\mathfrak g}\) be a Kac-Moody Lie algebra over a field of characteristic zero, and G its adjoint group. The algebra \({\mathfrak g}\) has a natural \({\mathbb{Z}}\)-grading, which allows us to construct the completed Lie algebra \(\hat {\mathfrak g}\) and the completed adjoint group \(\hat G\) associated with (\({\mathfrak g},G)\). Recently R. Moody has established that \(\hat G\) is an abstract simple group if \({\mathfrak g}\) is not of Euclidean \((=affine)\) type. The main result of this paper is that \(\hat G\) is also simple in case of Euclidean type, which comes from the fact that \(\hat G\) is isomorphic to a formal or twisted adjoint Chevalley group over a field of formal power series.

MSC:

20E32 Simple groups
20F40 Associated Lie structures for groups
17B65 Infinite-dimensional Lie (super)algebras
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References:

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