# Generalizations of Lie algebras

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.)

• Lie superalgebra is not the 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. – YCor Jan 27 '14 at 19:01

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.