# Graph of a Lie super algebra

Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\mathfrak{g}$ (At least in the affine case I know this is true).

My question is which Lie super algebras has an associated graph. If you can quote some reference regarding this, it will be very helpful to me.

Thank you.

Have a good day.

See §15 in Dictionary on Lie Superalgebras, by L. Frappat, A. Sciarrino, P. Sorba. They define the Dynkin diagram of basic Lie superalgebras (i.e., those with an even nondegenerate invariant bilinear form, and whose even part is reductive).

A good review article on Lie superalgebras by V.G. Kac can be found here: https://www.sciencedirect.com/science/article/pii/0001870877900172

Certain subtle points about Cartan matrices and Dynkin diagrams in the context of Lie superalgebras are clarified in the article "Serre presentations of Lie superalgebras" by R. Zhang which can be found here: https://link.springer.com/book/10.1007%2F978-3-319-02952-8

Classical, Simple, Complex, Lie superalgebras and Complex, Affine, Kac-Moody algebras and Complex, Kac-Moody Lie superalgebras have an associated graph -up to isomorphism- in the sense of a generalized Dynkin diagram :

Apart from the classical references (i.e. the articles of Kac and the dictionary on Lie superalgebras) which have already been mentioned in the preceding answers, an invaluable -imo- reference is the 3rd volume of J.F. Cornwell, Group Theory in Physics. Volume III: Supersymmetries and Infinite-Dimensional Algebras.

This book is maybe not so well known in the pure mathematics literature -it comes from the mathematical physics ... culture- but it contains an extreme wealth of detailed computations and examples, keeping at the same time a complete and rigorous presentation of the theory:
The classification of fin dim Classical, Simple, Complex, Lie superalgebras is presented in ch. 25 and detailed info on basis elements, dual spaces, roots, generalized dynkin diagrams, bilinear forms, matrix realizations etc is presented in Appendix M.
The same stuff but for Complex, Affine, Kac-Moody algebras and Complex, Kac-Moody Lie superalgebras is presented in ch. 26 and Appendix N. (however, the generalized dynkin diagrams are missing for the Kac-Moody Lie superalgebras case - there are just some hints and tips in p. 337).

• thanks. but you meant Complex Kac-Moody Lie superalgebras instead of Complex affine Kac-Moody algebras? – GA316 Sep 7 '18 at 3:50
• In fact, both Kac-Moody Lie algebras and Lie superalgebras are considered in J.F. Cornwell's book, ch. 26. – Konstantinos Kanakoglou Sep 7 '18 at 13:30