In this article we introduce the notion of the equivalence relation, isoclinism, on the central extensions of Lie algebras, and determine all central extensions occurring in an isoclinism class of a given central extension. We also show that under some conditions, the concepts of isoclinism and isomorphism between the central extensions of finite dimensional Lie algebras are identical. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 598 – 606 ¨ ITAK ˙ c TUB doi: Isoclinic extensions of Lie algebras Hamid MOHAMMADZADEH,1,∗ Ali Reza SALEMKAR,2 Zahra RIYAHI2 1 School of Mathematics, Iran University of Sciences and Technology, Tehran, Iran 2 Faculty of Mathematical Sciences, Shahid Beheshti University, ., Tehran, Iran Received: • Accepted: • Published Online: • Printed: Abstract: In this article we introduce the notion of the equivalence relation, isoclinism, on the central extensions of Lie algebras, and determine all central extensions occurring in an isoclinism class of a given central extension. We also show that under some conditions, the concepts of isoclinism and isomorphism between the central extensions of finite dimensional Lie algebras are identical. Finally, the connection between isoclinic extensions and the Schur multiplier of Lie algebras are discussed. Key words: Lie algebra, isoclinic extensions, Schur multiplier, stem cover 1. Introduction In 1940, P. Hall [6] introduced an equivalence relation on the class of all groups called isoclinism, which is weaker than isomorphism and plays an important role in classification of finite p-groups. This notion has since been further studied by a number of authors, including Bioch [4], Hekster [7], Jones and Wiegold [8], and Weichsel [16]. In 1994, K. Moneyhun [10] gave a Lie algebra analogue of isoclinism as follows: Two Lie algebras L1 and L2 are isoclinic if there exists an isomophism γ between the central quotients L1 /Z(L1 ) and L2 /Z(L2 ) and an isomorphism β between the derived subalgebras L21 and L22 such that γ and β are compatible with the commutator maps of L1 and L2 . Evidently, this produces a partition on the class of all Lie algebras into equivalence classes, the so-called isoclinism families. Note that the class of all abelian Lie algebras, .
