Since a Lie algebra is a vector space, $\Pi\mathfrak{g}$ is just a the vector space $\mathfrak{g}$ with a $\mathbb{Z}_2$ grading where every element is odd. In particular, if $t_a$ is a basis of $\mathfrak{g}$ and $c^a$ is the dual basis, then we can take $c^a$ as coordinates on $\Pi\mathfrak{g}$. Then functions on $\Pi\mathfrak{g}$ are polynomials in the c variables where we identify $c^ac^b$ with $-c^bc^a$. A good resource for these matters is https://arxiv.org/pdf/math/0402057.pdf
Ivan Burbano
- 123
- 7