A type Q Kac-Moody construction

Abstract

We introduce a new, Kac--Moody-flavoured construction for Lie superalgebras, which incorporates phenomena of the type Q (queer) Lie superalgebra. This is done by replacing a maximal even torus by the most general possible Cartan subalgebra for Lie superalgebras, which is a maximal quasitoral subalgebra. The theory is remarkably rigid but nevertheless unveils a new natural class of Lie superalgebras, which we call type Q Kac--Moody (QKM) algebras. We classify finite-growth type Q Kac--Moody algebras, and obtain in a novel way the $d=2$, $\mathcal{N}=1,2,3,4$ twisted superconformal algebras, along with three other new, finite growth Lie superalgebras. Our work also gives a new perspective on the distinctiveness of the Lie superalgebra $\mathfrak{q}(n)$.