부산대학교 도서관

A Lie algebra $T$ with ideal $G$ can be considered as a crossed module, and as such one can consider its central extensions in the category of crossed modules. If the ideal $G$ satisfies $G=[T,G]$ then there exists a universal central extension $(\widetilde T,\widetilde G)$ of the crossed module $(T,G)$. (In fact $\widetilde T=T\otimes G$ and $\widetilde G=G\otimes G$, where $\otimes$ denotes the tensor product discussed earlier by the reviewer [Glasgow Math. J. {\bf 33} (1991), no.~1, 101--120; MR1089961 (92g:18010)].) In the paper under review it is shown that the kernel of the homomorphism $\widetilde T\to\widetilde G$ is precisely the second homology invariant of the crossed module $(T,G)$ that was introduced by the author and M. Ladra [Bull. Soc. Math. Belg. Sér. A {\bf 45} (1993), no.~1-2, 59--84; MR1316232 (96a:17014)]. Furthermore, a criterion is given for recognizing precisely when a central extension of crossed modules is universal.