Strictly Imaginary Roots and Purely Imaginary Roots of BKM Lie Superalgebras

Abstract

In this article, we extend the concepts of strictly imaginary roots and purely imaginary roots of Borcherds Kac–Moody algebras (BKM algebras) to BKM superalgebras. We give a complete classification of those BKM superalgebras with purely alien imaginary property and purely imaginary property. We define a new class of indefinite non-hyperbolic BKM superalgebras called extended-hyperbolic BKM superalgebras and find that it always possesses purely alien imaginary property, but it does not always possess the purely imaginary property. Moreover, the properties of strictly imaginary roots and purely imaginary roots of BKM superalgebras were compared and the BKM superalgebras whose purely imaginary roots are also strictly imaginary were found out.