Classification of simple Lie superalgebras

Remarks

The theorem is due to Kac, in 1977. The families in items 1 to 3 are the classical simple Lie superalgebras: those for which $\mathfrak{g}_{\bar 0}$ is a reductive Lie algebra acting completely reducibly on $\mathfrak{g}_{\bar 1}$. The families in item 4 are the simple ones for which this fails. The proof treats the two cases separately. In the classical case it studies the representation of the reductive Lie algebra $\mathfrak{g}_{\bar 0}$ on $\mathfrak{g}_{\bar 1}$. In the other case it studies a filtration of $\mathfrak{g}$ and its associated graded superalgebra.

Among the classical ones, the series $A$ to $D$ and the exceptional algebras carry a nondegenerate even invariant bilinear form, while $P(n)$ and $Q(n)$ do not. For the algebras with such a form, there is a theory of root systems and Dynkin diagrams parallel to the classification of simple Lie algebras. A new feature is that a Lie superalgebra may have several inequivalent sets of simple roots.

Going deeper

For a student

The superalgebras $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(m|2n)$ are the direct analogues of the special linear and orthogonal or symplectic Lie algebras. In the Cartan type series, $W(n)$ is the Lie superalgebra of all superderivations of the exterior algebra $\Lambda(n)$. The series $S(n)$, $\widetilde{S}(n)$ and $H(n)$ are analogues of divergence-free and Hamiltonian vector fields. Unlike the Lie algebra case, finite-dimensional representations of simple Lie superalgebras need not be completely reducible.