The weak commutativity construction for Lie algebras

JOURNAL OF ALGEBRA

卷:529

Article

de Mendonca, Luis Augusto¹ 

[1] Univ Campinas UNICAMP, Dept Math, Rua Sergio Buargue de Holanda 651, BR-13083859 Campinas, SP, Brazil

关键词: Lie algebras;    Homological finiteness properties;    Finite presentability;    Schur multiplier;   

DOI  :  10.1016/j.jalgebra.2019.04.007

来源: Elsevier

【 摘 要 】

We study the analogue of Sidki's weak commutativity construction, defined originally for groups, in the category of Lie algebras. This is the quotient chi(g) of the Lie algebra freely generated by two isomorphic copies g and g(psi) of a fixed Lie algebra by the ideal generated by the brackets [x,x(psi)] for all x. We exhibit an abelian ideal of chi(g) whose associated quotient is a subdirect sum in g circle plus g circle plus g and we give conditions for this ideal to be finite dimensional. We show that chi(g) has a sub quotient that is isomorphic to the Schur multiplier of g. We prove that chi(g) is finitely presentable or of homological type FP2 if and only if g has the same property, but chi(f) is not of type FP3 if f is a non-abelian free Lie algebra. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free