Generators and relations for (generalised) Cartan type superalgebras

Abstract

In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, Apn´1, 0q " slp1|nq can be constructed by adding a "gray" node to the Dynkin diagram of An´1 " slpnq, corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is W pnq, the derivation algebra of the Grassmann algebra on n generators. Here we present a novel construction of W pnq, from the same Dynkin diagram as Apn´1, 0q, but with additional generators and relations.

L ∞ algebras for extended geometry

“…[50] we presented a set of generators and relations for the (finite-dimensional) tensor hierarchy algebras W (r + 1) = W (A r ) and S(r + 1) = S(A r ), based on the same Dynkin diagram as that of B(A r ). The straightforward generalisation of these relations seems to provide a good definition of W (g) and S(g) in general (see the talk by JP at the present meeting [52]). The superalgebra used for extended geometry with structure algebra g is S(g r+1 ).…”