nLab
n-Lie algebra

Idea

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

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

• but without necessarily the grading underlying an $L_{\mathrm{\infty }}$-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 $L_{\mathrm{\infty }}$-algebras.

So $n$-Lie algebras in this sense in general don’t integrate to Lie infinity-groupoids via the usual Lie theory. An important application of $n$-Lie algebras is as formalizations and generalizations of Nambu brackets.

References

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

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

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

Similar (but different) discussion is in

• P. Hanlon, M. Wachs, On Lie k-Algebras, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236

• J. A. de Azcarraga, J. C. Perez Bueno, Higher-order simple Lie algebras, (arXiv:hep-th/9605213)

Re-inventions

The notion of $n$-Lie algebras, for $n=3$, was re-invented 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 $L_{\mathrm{\infty }}$-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)