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There is a classification of simple Lie algebras in $\text{Vec}_{\mathbb{C}}$ given by Dynkin diagrams. We then have 4 families of simple lie algebras, plus some exceptional ones.

Question: How about simple Lie algebras in the bigger category $\text{sVec}_{\mathbb{C}}$ of super vector spaces? Is there a known classification for simple Lie algebras there?

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Yes there is a complete classification of finite dimensional, simple Lie superalgebras (over $\mathbb{C}$), which -up to a certain extent- goes very much in parallel with the corresponding case of Lie algebras and incorporates the later as a special case. There are significant conceptual differences though (as to the role and uniqueness of Dynkin diagrams, Cartan matrices etc). Historically it has been fully developed by Kac. The original references are:

  • V.G. Kac, Lie Superalgebras, Adv. Math. 26 (1977) 8.
  • V.G. Kac, A sketch of Lie superalgebra theory, Commun. Math. Phys. 53 (1977) 31.
  • V.G. Kac, Representations of classical Lie superalgebras, Lectures Notes in Mathematics 676 (1978) 597; Springer-Verlag, Berlin.

The above provide a self-contained full account of the work. You can also find a succint description of the classification with lots of detailed references at: the Dictionary of Lie superalgebras, arXiv:hep-th/9607161v1.

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