An -Lie algebra is defined to be an algebraic structure which
looks formally like the special case of an -algebra for which only the -ary bracket is non-vanishing (see there);
but without necessarily the grading underlying an -algebra, and in particular without the requirement that be of homogeneous degree in any grading.
Therefore, any “-Lie algebras” that appear in the literature are not examples of Lie n-algebras, hence of L-∞ algebras. (So in particular -Lie algebras in this sense in general don’t integrate to Lie infinity-groupoids via the usual Lie theory. )
A discussion of -Lie algebras (without the -grading) is in
v. T. Filippov, -Lie algebras, Sib. Math. Zh. No 6 126–140 (195)
v. T. Filippov, On the -Lie algebra of Jacobians, Sibirsk. Mat. Zh., 1998, Volume 39, Number 3, Pages 660–669 (English translation)
A. S. Dzhumadil’daev, Wronskians as -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
The notion of -Lie algebras, for , 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 -algebraic formalism.
Further re-inventions of the concept of -Lie algebras in this context are appearing. For instance in
based on a result in
and reviewed in