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

Consider a Borcherds-Kac-Moody Lie superalgebra, denoted as $g$, associated with the graph G. This Lie superalgebra is constructed from a free Lie superalgebra by introducing three sets of relations on its generators: (1) Chevalley relations, (2) Serre relations, and (3) The commutation relations derived from the graph G.

The Chevalley relations lead to a triangular decomposition of $g$ as $g=n_{+}\oplus h\oplus n_{-}$, where each root space $g_{\alpha }$ is contained in either $n_{+}$ or $n_{-}$. Importantly, each $g_{\alpha }$ is determined solely by relations (2) and (3). This paper focuses on the root spaces of $g$ that are unaffected by the Serre relations. We refer to these root spaces as “free roots” of $g$ (these root spaces are free from the Serre relations and can be associated with certain grade spaces of freely partially commutative Lie superalgebras, as detailed in Lemma 3.10. Consequently, we refer to them as “free roots,” and the corresponding root spaces in $g$ as “free root spaces” [cf. Definition 2.6]). Since these root spaces only involve commutation relations derived from the graph G, we can examine them purely from a combinatorial perspective.

We employ heaps of pieces to analyze these root spaces and establish various combinatorial properties. We develop two distinct bases for these root spaces of $g$: We extend Lalonde's Lyndon heap basis, originally designed for free partially commutative Lie algebras, to accommodate free partially commutative Lie superalgebras. We expand upon the basis introduced in the reference [1], designed for the free root spaces of Borcherds algebras, to encompass BKM superalgebras. This extension is achieved by investigating the combinatorial properties of super Lyndon heaps. Additionally, we also explore several other combinatorial properties related to free roots.