I am reading the paper “Kac–Moody superalgebras and integrability” by Serganova (2011), which is highly relevant to my ongoing research. Properly understanding real roots of a Kac–Moody superalgebra $\mathfrak{g}$ (see Definition 4.1 of the paper) feels somewhat enigmatic to me. I've tried to understand a few specific questions but haven't fully resolved them:
1. Symmetry of real roots:
Is the set of real roots (defined just after Corollary 4.10) closed under negation?
- For a regular algebra $\mathfrak{g}$ (Definition 3.4), this is true.
- But for the non-regular case, I'm not even sure whether $-\Pi$ (the set of negatives of standard simple roots) is a subset of the set of real roots.
2. Addition with a simple isotropic root:
Let $\beta$ be a real root and let $\alpha_i$ be a standard simple isotropic root. When is $\beta + \alpha_i$ a real root?
- For regular $\mathfrak{g}$, I am only able to prove (the trivial case) that $\alpha_i + \alpha_j$ is a real root for another standard simple root $\alpha_j$ if and only if $\alpha_j(h_{\alpha_i}) \neq 0$.
- However, in the non-regular case, I couldn't prove even this.
3. Root strings with isotropic real roots:
I also want to understand the root string $\{\beta + k\alpha : k \in \mathbb{Z}\}$ when $\alpha, \beta$ are real roots, and $\alpha$ is isotropic.
- In finite-dimensional and affine cases, this string contains at most two roots.
- But I am unclear what happens in general for infinite-dimensional (non-affine) quasisimple cases.
4. Description of imaginary roots:
Is there any (more practically workable) description of imaginary roots beyond Corollary 4.12?
Any suggestions, comments, or remarks would be highly appreciated. Thank you!
