Pairs of Finite Dimensional Nilpotent and Filiform Lie Algebras

Document Type : Original Research Article

Authors

1 Department of Mathematics, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran.

2 Department of Mathematics, Shahr-e-Qods Branch, Islamic Azad University Tehran, Iran;

Abstract

Let (N,L) be a pair of finite dimensional nilpotent Lie algebras. If N admits a complement K in L such that dim N = n and dim K = m, then dim M(N,L) = 1/2n(n + 2m - 1) - t(N,L), where M(N,L) is the Schur multiplier of the pair (N,L) and t(N,L) is a non-negative integer. In this paper, we characterize the pair (N,L) for t(N,L)=0, 1, 2, , 23, where N is a finite dimensional filiform Lie algebra and N,K are ideals of L such that L = N ⊕ K. Moreover, we classify the pair (N,L) for s(N,L) = 3, where S(N,L) = 1/2 (n - 1)(n - 2) + 1 + (n - 1)m dim M(N,L), L is a finite dimensional nilpotent Lie algebra and N is a non-abelian ideal of L.


Keywords


  1. H. Arabyani, F. Saeedi, M. R. R. Moghaddam and E. Khamseh, Characterization of nilpotent Lie algebras pair by their Schur multipliers, Commun. Algebra, 42, 5474–5483 (2014).
  2. P. Batten, K. Moneyhun, E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, Commun. Algebra, 24, 4319–4330 (1996).
  3. G. Ellis, The Schur multiplier of a pair of groups, Appl. Categ. Struct., 6, 355–371 (1998).
  4. J. R. Gomez, A. Jimenez-Merchan, Y. Khakimdjanov, Low-dimensional filiform Lie algebras, J. Pure Appl. Algebra, 130, 133–158 (1998).
  5. P. Hardy, On characterizing nilpotent Lie algebras by their multipliers III, Commun. Algebra, 33, 4205–4210 (2005).
  6. P. Hardy, E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, t(L) = 3, 4, 5, 6, Commun. Algebra, 26, 3527–3539 (1998).
  7. E. Khamseh, S. A. Niri, Classification of pair of nilpotent Lie algebras by their Schur multipliers, Math. Reports, 20, 177–185(2018).
  8. K. Moneyhun, Isoclinisms in Lie algebras, Algebras Groups Geom., 11, 9–22 (1994).
  9. P. Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras, Cent. Eur. J. Math, 9, 57–64 (2011).
  10. P. Niroomand, F. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Commun. Algebra, 39, 1293–1297 (2011).
  11. F. Saeedi, H. Arabyani, P. Niroomand, On dimension of Schur multiplier of nilpotent Lie algebras II, Asian-Eur. J. Math, 10, 1750076 (8 pages) (2017).
  12. F. Saeedi, A. R. Salemkar, B. Edalatzadeh, The commutator subalgebra and Schur multiplier of a pair of nilpotent Lie algebras, J. Lie Theory, 21, 491–498 (2011).
  13. A. Shamsaki, P. Niroomand, The Schur multipliers of Lie algebras of maximal class, Internat. J. Algebra. Comput, 29, 795–801 (2019).
Volume 6, Issue 2
November 2021
Pages 179-186
  • Receive Date: 27 September 2021
  • Revise Date: 11 November 2021
  • Accept Date: 22 November 2021
  • Publish Date: 01 November 2021