I often stumble over "Lie superalgebra" (= "Lie algebra with a Z2 grading"). Obvious question: What about Z3 grading (and so on)? Is a Lie algebra with Zn grading just the special case of a quantum Lie algebra L(q) with q being a 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*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.)

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.)