# nLab n-Lie algebra

-Lie algebras

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).

# $n$-Lie algebras

## 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).

## 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

### 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

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

• 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

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