Involve: A Journal of Mathematics

  • Involve
  • Volume 9, Number 5 (2016), 783-795.

The left greedy Lie algebra basis and star graphs

Benjamin Walter and Aminreza Shiri

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We construct a basis for free Lie algebras via a left greedy bracketing algorithm on Lyndon–Shirshov words. We use a new tool — the configuration pairing between Lie brackets and graphs of Sinha and Walter — to show that the left greedy brackets form a basis. Our constructions further equip the left greedy brackets with a dual monomial Lie coalgebra basis of star graphs. We end with a brief example using the dual basis of star graphs in a Lie algebra computation.

Article information

Involve, Volume 9, Number 5 (2016), 783-795.

Received: 29 April 2015
Revised: 23 October 2015
Accepted: 27 October 2015
First available in Project Euclid: 22 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B35: Universal enveloping (super)algebras [See also 16S30]
Secondary: 17B62: Lie bialgebras; Lie coalgebras 16T15: Coalgebras and comodules; corings 18D50: Operads [See also 55P48]

Lie algebra bases free Lie algebra


Walter, Benjamin; Shiri, Aminreza. The left greedy Lie algebra basis and star graphs. Involve 9 (2016), no. 5, 783--795. doi:10.2140/involve.2016.9.783.

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