Involve: A Journal of Mathematics

  • Involve
  • Volume 7, Number 5 (2014), 647-655.

A not-so-simple Lie bracket expansion

Julie Beier and McCabe Olsen

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Lie algebras and quantum groups are not usually studied by an undergraduate. However, in the study of these structures, there are interesting questions that are easily accessible to an upper-level undergraduate. Here we look at the expansion of a nested set of brackets that appears in relations presented in a paper of Lum on toroidal algebras. We illuminate certain terms that must be in the expansion, providing a partial answer for the closed form.

Article information

Involve, Volume 7, Number 5 (2014), 647-655.

Received: 14 May 2013
Revised: 3 February 2014
Accepted: 3 March 2014
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Lie algebra toroidal algebra


Beier, Julie; Olsen, McCabe. A not-so-simple Lie bracket expansion. Involve 7 (2014), no. 5, 647--655. doi:10.2140/involve.2014.7.647.

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  • A. Borel, Essays in the history of Lie groups and algebraic groups, History of Mathematics 21, American Math. Soc. and London Math. Soc., 2001.
  • K. H. Lum, “A presentation of toroidal algebras as homomorphic images of G.I.M. algebras”, Comm. Algebra 26:12 (1998), 4051–4063.
  • G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Birkhäuser, Boston, 1993.