Journal of Lie Theory Volume 6 (1996) 207-213 C ◯1996 Heldermann Verlag

A Note on Central Extensions of Lie Groups

Karl-Hermann Neeb

Communicated by K. H. Hofmann

Abstract. In this note we show that the existence of a central extension of a Lie groups G for a prescribed central extension of its Lie algebra can be completely characterized by the exactness of a certain set of 1-forms on G which are obtained by contracting the left-invariant 2-form Ω defined by the 2-cocycle of the Lie algebra extension with right-invariant vector fields. This criterion simplifies a criterion derived by Tuynman and Wiegerinck in the sense that they required in addition that the group of periods of Ω is discrete.

Karl-Hermann Neeb Mathematisches Institut Universität Erlangen-Nürnberg Bismarckstr. 1 1/2 D-91054 Erlangen Germany

Received May 30, 1996

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag