Research ArticleNo Access

On the combinatorics of commutators of Lie algebras

Eduardo Eizo Aramaki Hitomi

Department of Mathematics, State University of Campinas, Campinas, SP 13083-859, Brazil

E-mail Address: ehitomi@ime.unicamp.br

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and

Felipe Yukihide Yasumura

http://orcid.org/0000-0001-6623-6874

Department of Mathematics, State University of Maringá, Maringá, PR 87020-900, Brazil

E-mail Address: felipeyukihide@gmail.com

Corresponding author.

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https://doi.org/10.1142/S0219498820501194Cited by:1 (Source: Crossref)

Abstract

Motivated by the combinatorial properties of products in Lie algebras, we investigate the subset of permutations that naturally appears when we write the long commutator $[x_{1}, x_{2}, \dots, x_{m}]$ $[x_{1}, x_{2}, \dots, x_{m}]$ as a sum of associative monomials. We characterize this subset and find some useful equivalences. Moreover, we explore properties concerning the action of this subset on sequences of $m$ $m$ elements. In particular, we describe sequences that share some special symmetries which can be useful in the study of combinatorial properties in graded Lie algebras.

Communicated by L. H. Rowen

Keywords:

AMSC: 17B05, 05E18

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