TAILIEUCHUNG - Some upper bounds on the dimension of the Schur multiplier of a pair of nilpotent Lie algebras

Let (L, N) be a pair of Lie algebras where N is an ideal of the finite dimensional nilpotent Lie algebra L. Some upper bounds on the dimension of the Schur multiplier of (L, N) are obtained without considering the existence of a complement for N . These results are applied to derive a new bound on the dimension of the Schur multiplier of a nilpotent Lie algebra. | Turk J Math (2016) 40: 1020 – 1024 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Some upper bounds on the dimension of the Schur multiplier of a pair of nilpotent Lie algebras Behrouz EDALATZADEH∗ Department of Mathematics, Faculty of Science, Razi University, Kermanshah, Iran Received: • Accepted/Published Online: • Final Version: Abstract: Let (L, N ) be a pair of Lie algebras where N is an ideal of the finite dimensional nilpotent Lie algebra L . Some upper bounds on the dimension of the Schur multiplier of (L, N ) are obtained without considering the existence of a complement for N . These results are applied to derive a new bound on the dimension of the Schur multiplier of a nilpotent Lie algebra. Key words: Pair of Lie algebras, Schur multiplier, nilpotent Lie algebra 1. Introduction Throughout this paper, we denote by (L, N ) a pair of Lie algebras where N is an ideal of the Lie algebra L . The Schur multiplier of the pair (L, N ) is defined to be the abelian Lie algebra M(L, N ), whose principal feature is the following natural exact sequence of Lie algebras: H3 (L) → H3 (L/N ) → M(L, N ) → H2 (L) → H2 (L/N ) → N/[N, L] → H1 (L) → H1 (L/N ) → 0, (1) where Hi (−) is the i-th Chevalley–Eilenberg homology group of a Lie algebra. From the homotopical point of view, M(L, N ) is the second relative homology of (L, N ), see [3, 4] for more details and a brief introduction. Taking N = L we find that M(L, N ) = H2 (L) , which is called the Schur multiplier of L and denoted by M(L). Determining bounds on the dimension of the Schur multiplier of a (nilpotent) Lie algebra was a hot topic in recent decades. Nilpotent Lie algebras have been widely discussed in the literature in order to be classified by their multipliers; however, there are many other interesting open problems on the dimension of the homology groups of nilpotent Lie algebras; see .

• Toán học
• Pair of Lie algebras
• Schur multiplier
• Nilpotent Lie algebra
• Chevalley Eilenberg homology group
• Homotopical point