I often stumble over the term "Lie superalgebra" (= "Lie algebra with a $\mathbb{Z}_2$ grading"). Obvious question: What about $\mathbb{Z}_3$ grading (and so on)? Is a Lie algebra with $\mathbb{Z}_n$ grading just the special case of a quantum Lie algebra $L(q)$ with $q$ being an $n$-th root of 1 (I only looked at the commutator equation :-) or are these completely different things?
And are there other generalizations of Lie algebras I should know? (Just to get concrete, what is the Lie algebra series behind the "Vogel plane" for a thing?)

(Sidenote: I'm also asking because I found a very special tangled graph invariant which doesn't differ from any "standard" Reshitikhine-Turaev invariant in any relevant property, but if you look closely, the adjoint splits as $6 \cdot 6=1+1+6+8+8+12$ and the metric tensor is not singular. The latter rules out non-semisimple Lie algebras and the $1+1$ semisimple ones. So my first thought was it might come from a Lie superalgebra.)

Such generalizations exist. It is well known that the classical
Clifford algebras can be used to construct Lie superalgebras. The main
tool of the construction is the notion of the $\mathbb{Z}_2$-graded commutator which includes both the ordinary commutator and anticommutator. This can be generalized. For the case of ternary $\mathbb{Z}_3$-graded commutators via ternary analogues of Clifford algebras see for example the paper of *Viktor Abramov*: "$\mathbb{Z}_3$-graded analogues of Clifford
algebras and generalization of supersymmetry"

A generalization of the idea of a Lie superalgebra exists and is known as $\epsilon$-Lie algebras or color Lie algebras. They generalize Lie superalgebras in the sense that the underlying vector space is $G$-graded instead of $\mathbb{Z}_{2}$, where $G$ is an arbitrary abelian group. Of course, one modifies the skew-symmetry and skew Jacobi identities, using commutation factors.

See for example "Scheunert, M. Generalized Lie algebras. J. Math. Phys. 20 (1979), no. 4, 712–720."

notthe same as a Lie algebra with a $\mathbb{Z}/2\mathbb{Z}$-grading (which is also something useful but does not require a special name), because there is a twisting of the Lie algebra axioms as well. See en.wikipedia.org/wiki/Lie_superalgebra. $\endgroup$ – YCor Jan 27 '14 at 19:01