PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 1, January 2014, Pages 121–127 S 0002-9939(2013)11776-4 Article electronically published on October 3, 2013 INEQUALITIES FOR THE SECOND COHOMOLOGY OF FINITE DIMENSIONAL LIE ALGEBRAS ALI REZA SALEMKAR, BEHROUZ EDALATZADEH, AND HAMID MOHAMMADZADEH (Communicated by Kailash C. Misra) Abstract. We will extend the Hochschild-Serre spectral sequence for coho- mology of Lie algebras a step further. Also, some inequalities and upper bounds for the dimension of the second cohomology of finite dimensional Lie algebras will be given. 1. Introduction All Lie algebras are considered over a fixed field Λ and [ , ] denotes the Lie bracket. Let L be a Lie algebra, K an ideal of L and M an L-module. Let H n (L, M ) and H n (L, M ) denote the nth homology and nth cohomology of L with coefficients in M , respectively. Then the Hochshild-Serre spectral sequence for homology and cohomology of Lie algebras yields the following exact sequences: (1) H 2 (L, M )−→H 2 ( L K ,M K )−→H 1 (K, M ) L/K −→H 1 (L, M )−→H 1 ( L K ,M K )−→0, (2) 0−→H 1 ( L K ,M K )−→H 1 (L, M )−→H 1 (K, M ) L/K −→H 2 ( L K ,M K )−→H 2 (L, M ). Here M K and M K are the invariant and the coinvariant submodules of M , respec- tively, in which M is regarded as an L/K-module [10]. Consider the field Λ as a trivial L-module. In 1991, Ellis [4] showed that the exact sequence (1) can be extended to an infinite long exact sequence. In particular, he obtained the following exact sequence: H 2 (L; K)−→H 2 (L, Λ)−→H 2 ( L K , Λ)−→H 1 (L; K)−→H 1 (L, Λ)−→H 1 ( L K , Λ)−→0, where H 1 (L; K) ∼ = L/[L, K] and H 2 (L; K) is the kernel of the commutator map L ∧ K−→L (here L ∧ K denotes the non-abelian exterior product of Lie algebras [3]). Received by the editors July 31, 2011 and, in revised form, March 12, 2012. 2010 Mathematics Subject Classification. Primary 17B30, 17B56. Key words and phrases. Lie algebra, cohomology group. This research was supported by a grant from Shahid Beheshti University. c 2013 American Mathematical Society 121 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use