Borcherds–Bozec algebras, root multiplicities and the Schofield construction

Using the twisted denominator identity, we derive a closed form root multiplicity formula for all symmetrizable Borcherds–Bozec algebras and discuss its applications including the case of Monster Borcherds–Bozec algebra. In the second half of the paper, we provide the Schofield construction of symmetric Borcherds–Bozec algebras.

Peterson-type dimension formulas for graded Lie superalgebras

Let be a free abelian group of finite rank, let Γ be a sub-semigroup of satisfying certain finiteness conditions, and let be a (Γ × Z2)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of , we derive a new recursive formula for the dimensions of homogeneous subspaces of . When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson’s root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.