A study on free roots of Borcherds-Kac-Moody Lie Superalgebras

Abstract

Let $\mathfrak{g}$ be a Borcherds-Kac-Moody Lie superalgebra (BKM superalgebra in short) with the associated graph $G$. Any such $\mathfrak{g}$ is constructed from a free Lie superalgebra by introducing three different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, and (3) Commutation relations coming from the graph $G$. By Chevalley relations we get a triangular decomposition $\mathfrak{g}={\mathfrak{n}}_{+}\oplus \mathfrak{h}\oplus {\mathfrak{n}}_{-}$ and each roots space ${\mathfrak{g}}_{\hat{\mathfrak{I}}}\pm$ is either contained in ${\mathfrak{n}}_{+}$ or ${\mathfrak{n}}_{-}$. In particular, each ${\mathfrak{g}}_{\hat{\mathfrak{I}}}\pm$ involves only the relations (2) and (3). In this paper, we are interested in the root spaces of $\mathfrak{g}$ which are independent of the Serre relations. We call these roots free roots${}^1$ of $\mathfrak{g}$. Since these root spaces involve only commutation relations coming from the graph $G$ we can study them combinatorially. We use heaps of pieces to study these roots and prove many combinatorial properties. We construct two different bases for these root spaces of $\mathfrak{g}$: One by extending the Lalonde's Lyndon heap basis of free partially commutative Lie algebras to the case of free partially commutative Lie superalgebras and the other by extending the basis given in \cite{akv17} for the free root spaces of Borcherds algebras to the case of BKM superalgebras. This is done by studying the combinatorial properties of super Lyndon heaps. We also discuss a few other combinatorial properties of free roots.