Graded embeddings, root generated subalgebras and $\pi$-systems for quasisimple Kac-Moody superalgebras

TL;DR

This paper explores the structure of roots and subalgebras in quasisimple Kac-Moody superalgebras, establishing new bijections and properties of root systems through reflections and $pi$-systems.

Contribution

It introduces a new understanding of real roots via reflections, establishes an analogue of Dynkin's bijection, and characterizes graded embeddings in the superalgebra context.

Findings

Real roots of subalgebras are obtained by reflections.

An analogue of Dynkin's bijection is established.

Graded embeddings correspond to linearly independent $pi$-systems.

Abstract

Motivated by a construction of Gorelik and Shaviv, we show that the real roots of a root generated subalgebra associated with a $\pi$-system contained in the positive roots are obtained by successive applications of even and odd reflections to the $\pi$-system, and that they form a real closed subroot system. Using this result, we establish an analogue of Dynkins bijection in the setting of symmetrizable quasisimple Kac-Moody superalgebras. In addition, we obtain several results on root strings in the super setting, analogous to those of Billig and Pianzola, and show that graded embeddings arise as root generated subalgebras associated with linearly independent $\pi$-systems.