∞-Lie theory (higher geometry)
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).
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
P. Hanlon, M. Wachs, On Lie k-Algebras, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236
José de Azcárraga, J. C. Perez Bueno, Higher-order simple Lie algebras, (arXiv:hep-th/9605213)
The notion of $n$-Lie algebras, for $n=3$, was re-invented by string physicists in the BLG model
which sparked a tremendous amount of activity.
See the blog entry
for further details and links. And see this blog discussion
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
An interpretation of $3$-Lie algebras as Lie 2-algebras equipped with a binary invariant polynomial (“metric Lie 2-algebras”) is due to
based on a result in
and reviewed in
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)
See also