nLab n-Lie algebra

This entry is about the concept that became famous with the BLG model while it is NOT about what in homotopy theory are known as Lie n-algebras (homotopy-theoretic Lie algebras), NOR n-algebras (categorifies associative algebras). Some authors also say Filippov algebra.

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Idea

An $n$-Lie algebra is defined to be an algebraic structure which

• looks formally like the special case of an $L_\infty$-algebra for which only the $n$-ary bracket $D_n$ is non-vanishing (see there);

• but without necessarily the grading underlying an $L_\infty$-algebra, and in particular without the requirement that $D_n$ be of homogeneous degree $-1$ in any grading.

Therefore, any “$n$-Lie algebras” that appear in the literature are not examples of Lie n-algebras, hence of L-∞ algebras. (So in particular $n$-Lie algebras in this sense in general don’t integrate to Lie infinity-groupoids via the usual Lie theory. )

Instead, at least certain “3-Lie algebras” can be understood as encoding structure in Lie 2-algebras equipped with a binary invariant polynomial (Saemann-Ritter 13, section 2.5).

Related concepts

• higher Poisson bracket

  • Nambu bracket

• membrane matrix model

• BLG model

References

Original

A discussion of $n$-Lie algebras (without the $L_\infty$-grading) is in

• V. T. Filippov, $n$-Lie algebras, Sib. Math. Zh. No 6 126–140 (195)

• V. T. Filippov, On the $n$-Lie algebra of Jacobians, Sibirsk. Mat. Zh., 1998, Volume 39, Number 3, Pages 660–669 (English translation)

• A. S. Dzhumadil’daev, Wronskians as $n$-Lie multiplications (arXiv:math/0202043)

Similar (but different) discussion is in

• Phil Hanlon, Michelle Wachs, On Lie $k$-Algebras, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236 (doi:10.1006/aima.1995.1038)

• José de Azcárraga, J. C. Perez Bueno, Higher-order simple Lie algebras, Commun. Math. Phys. 184 (1997) 669-681 [arXiv:hep-th/9605213, doi:10.1007/s002200050079]

• José de Azcárraga, José M. Izquierdo, $n$-Ary algebras: a review with applications, J. Phys. A: Math. Theor. 43 (2010) 293001 [doi:10.1088/1751-8113/43/29/293001]

Re-inventions

The notion of $n$-Lie algebras, for $n=3$, was re-invented, in the form of the M-brane 3-algebra by string physicists in the BLG model

• Jonathan Bagger, Neil Lambert, Gauge Symmetry and Supersymmetry of Multiple M2-Branes (arXiv)

which sparked a tremendous amount of activity.

See the blog entry

• Lie 3-Algebras on the Membrane

for further details and links. And see this blog discussion

• 3-Algebra and Chern-Simons terms

for discussion about the relation to proper $L_\infty$-algebraic formalism.

Further re-inventions of the concept of $n$-Lie algebras in this context are appearing. For instance in

• Tamar Friedman, Orbifold singularities, the LATKe, and Yang-Mills with Matter (arXiv)

Relation to representations of metric Lie algebras

The full generalized axioms on the M2-brane 3-algebra and first insights into their relation to Lie algebra representations of metric Lie algebras is due to

• Sergey Cherkis, Christian Saemann, Multiple M2-branes and Generalized 3-Lie algebras, Phys. Rev. D78:066019, 2008 (arXiv:0807.0808)

The full identification of M2-brane 3-algebras with dualizable Lie algebra representations over metric Lie algebras is due to

• Paul de Medeiros, José Figueroa-O'Farrill, Elena Méndez-Escobar, Patricia Ritter, On the Lie-algebraic origin of metric 3-algebras, Commun. Math. Phys. 290:871-902, 2009 (arXiv:0809.1086)

See also:

• Sam Palmer, Christian Saemann, section 2 of M-brane Models from Non-Abelian Gerbes, JHEP 1207:010, 2012 (arXiv:1203.5757)

• Patricia Ritter, Christian Saemann, section 2.5 of Lie 2-algebra models, JHEP 04 (2014) 066 (arXiv:1308.4892)

• Christian Saemann, appendix A of Lectures on Higher Structures in M-Theory (arXiv:1609.09815)

• José Figueroa-O'Farrill, Triple systems and Lie superalgebras in M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf, pdf)