Some Properties of the Schur Multiplier and Covers of Lie Algebras

In this article, we give the structure of all covers of Lie algebras that their Schur multipliers are finite dimensional, which generalizes the work of Batten and Stitzinger (1996 Batten , P. , Stitzinger , E. ( 1996 ). On covers of Lie algebras . Comm. Alg. 24 : 4301 – 4317 .Taylor & Francis Online, Web of Science ® , Google Scholar). Also, similar to a result of Yamazaki (1964 Yamazaki , K. ( 1964 ). Projective representations and ring extensions . J. Fac. Sci. Univ. Tokyo, Sect I 10 : 147 – 195 . Google Scholar) in the group case, it is shown that each stem extension of a finite dimensional Lie algebra is a homomorphic image of a stem cover for it. Moreover, we introduce an ideal in every Lie algebra, which is the smallest ideal contained in the center whose factor algebra is capable, and give some different forms of this ideal. Finally, we study the connection between this ideal and the concept of the Schur multiplier.