PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 144, Number 10, October 2016, Pages 4157–4168 http://dx.doi.org/10.1090/proc/13092 Article electronically published on May 6, 2016 ON THE CAPABILITY AND SCHUR MULTIPLIER OF NILPOTENT LIE ALGEBRA OF CLASS TWO PEYMAN NIROOMAND, FARANGIS JOHARI, AND MOHSEN PARVIZI (Communicated by Kailash C. Misra) Abstract. Recently, the authors in a joint paper obtained the structure of all capable nilpotent Lie algebras with derived subalgebra of dimension at most 1. This paper is devoted to characterizing all capable nilpotent Lie algebras of class two with derived subalgebra of dimension 2. It develops and generalizes the result due to Heineken for the group case. 1. Motivation and preliminaries According to Beyl and Tappe a p-group G is called capable if G is isomorphic to H/Z (H) for a group H. There are some fundamental known results concerning capability of p-groups. For instance, in [9, Corollary 4.16], it is shown the only capable extra-special p-groups (the p-group with Z (G)= G ′ and |G ′ | = p) are those of order p 3 and exponent p. In the case that G ′ = Z (G) and Z (G) is an elementary abelian p-group of rank 2, Heineken in [15] proved that the capable ones have order at most p 7 . Lie algebras and groups have similarities in the structures, so some authors tried to make analogies between them. But in this way not everything is the same and there are differences between groups and Lie algebras so that most of the time the proofs are different. Also the results in the field of Lie algebras are sometimes stronger than that for groups. For instance, in [23], the authors obtained the structure of a capable nilpotent Lie algebra L when dim L 2 ≤ 1. It developed the result of [9, Corollary 4.16] for groups to the case of Lie algebras. Recall that a Lie algebra is capable provided that L ∼ = H/Z (H) for a Lie algebra H. In the same area of research, we are going to characterize the structure of all capable Lie algebras that are nilpotent of class two with derived subalgebra of dimension 2. It obviously develops and generalizes the result of Heineken [15] for groups to the area of Lie algebras. As an application, we exactly obtain M(L), the Schur multiplier of those Lie algebras. Recall that if L is a Lie algebra and F a free Lie algebra such that L ∼ = F/R, then M(L), is isomorphic to R ∩ F 2 /[R, F ]. The reader can find some literatures about the Schur multiplier of groups and Lie algebras for instance in [1, 4, 5, 7, 9, 11, 19–29]. Throughout the paper, we assume that all Lie algebras have finite dimensions on an algebraically closed field, and we use the symbol H(m) for the Heisenberg algebra Received by the editors August 31, 2015 and, in revised form, December 21, 2015. 2010 Mathematics Subject Classification. Primary 17B30; Secondary 17B05, 17B99. Key words and phrases. Nilpotent Lie algebra, Schur multiplier, capable Lie algebra. The first author acknowledges the financial support of the research council of Damghan Uni- versity with the grant number 93/math/127/229. c 2016 American Mathematical Society 4157 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use