On the Canonical Construction of Simple Lie Superalgebras
J. Dhamothiran, Saudamini Nayak
TL;DR
This work addresses canonical construction of simple Lie superalgebras from abstract generalized root systems (GRS) with isotropic roots, leveraging Serganova's irreducible GRS classification. It develops a unified, Chevalley-type framework: a vector superspace $M$ with homogeneous generators $e_i,f_i,h_i$ acts to realize a Lie superalgebra $\mathfrak{g}$ whose root system is isomorphic to the original $R$, and a Cartan subalgebra $\mathfrak{h}$ yields a root-space decomposition. The paper proves $\mathfrak{g}$ is finite-dimensional and simple, with automorphisms relating $e_i$ and $f_i$, and it outlines a quiver-diagram approach to organize irreducible GRS by listing positive root data and bases for several families including $D(2,1;a)$, $G(3)$, $F(4)$ and the classical series. Overall, it provides a constructive, canonical bridge from abstract root systems to concrete simple Lie superalgebras, enriching the Kac–Moody-type framework for basic classical superalgebras and suggesting a unified geometric/combinatorial view via quivers.
Abstract
Axioms for the generalization of root systems were defined and classified (irreducible) by V. Serganova, which precisely correspond to the root systems of basic classical Lie Superalgebras. Here, we present a unified method for constructing simple Lie Superalgebras from the abstract root system, with the choice of base having the minimal number of isotropic roots.