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

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.