# Generators and relations for (generalised) Cartan type superalgebras

## TL;DR

This paper introduces a new construction method for Cartan superalgebras, specifically for $W(n)$, using Dynkin diagrams similar to those of contragredient Lie superalgebras but with added generators and relations.

## Contribution

It provides a novel presentation of $W(n)$, expanding the understanding of the structure of Cartan superalgebras through Dynkin diagram-based generators and relations.

## Key findings

- New construction of $W(n)$ from Dynkin diagrams
- Extended generators and relations for Cartan superalgebras
- Enhanced understanding of superalgebra structures

## 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, $A(n-1,0) = \mathfrak{sl}(1|n)$ can be constructed by adding a "gray" node to the Dynkin diagram of $A_{n-1} = \mathfrak{sl}(n)$, corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is $W(n)$, the derivation algebra of the Grassmann algebra on $n$ generators. Here we present a novel construction of $W(n)$, from the same Dynkin diagram as $A(n-1,0)$, but with additional generators and relations.