Rend. Circ. Mat. Palermo DOI 10.1007/s12215-016-0251-0 2-Nilpotent multipliers of a direct product of Lie algebras Peyman Niroomand 1,2 · Mohsen Parvizi 3 Received: 10 February 2016 / Accepted: 14 June 2016 © Springer-Verlag Italia 2016 Abstract In this paper, we present an explicit formula for the 2-nilpotent multiplier of a direct product of two Lie algebras. Keywords Schur multiplier · 2-Nilpotent multiplier · Free product · Free Lie algebra Mathematics Subject Classification 17b60 · 17b99 1 Introduction Throughout the paper [,] denotes the Lie bracket. Let L be a Lie algebra analogues to the theory of the group (see, for instance [5, 8, 10]) the c-nilpotent multiplier of L defined as a factor Lie algebra M (c) ( L ) = R ∩ F c+1 /( R, F ) c+1 , where F c+1 is the (c + 1)-th term of the derived series of F , ( R, F ) 1 = R, ( R, F ) c+1 = [( R, F ) c , F ] and L ∼ = F/ R for a free Lie algebra F . Similar to the group theory case, one may check that M (c) ( L ) is abelian and independent of the choice of the free Lie algebra F . In the case that c = 1, M (1) ( L ) is denoted by M( L ), The first authors research was in part supported by a grant from IPM (No. 94160063). B Peyman Niroomand p_niroomand@yahoo.com; niroomand@du.ac.ir Mohsen Parvizi parvizi@math.um.ac.ir 1 School of Mathematics and Computer Science, Damghan University, Damghan, Iran 2 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran 3 Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran 123