Abstract: Over 50 years ago, Victor Kac and Robert Moody introduced Kac-Moody algebras as a natural extension of the already classified semisimple Lie algebras. There are three types of Kac-Moody algebras: finite, affine, and indefinite. Both finite and affine have had all root multiplicities calculated. Some partial results have been obtained in the indefinite case, but the root multiplicities are not completely known. In this work, we realize the indefinite Kac-Moody algebras HE_7^(1)and HE_8^(1) as minimal graded Lie algebras whose local part is V\oplus gl(n;C)\oplus V' where V and V' are suitably chosen gl(n)-modules. This will allow us to use the combinatorial Kang's multiplicity formula to compute the root multiplicities to level 7 in the case of HE_7^(1) and level 9 in the case of HE_8^(1). Additionally, we verify the counterexample to Frenkel's conjecture for HE_8^(1) found by Kac, Moody, and Wakimoto by computing the relevant root multiplicity and we provide a root whose multiplicity is a counterexample to Frenkel's conjecture for HE_7^(1) , showing that Frenkel's conjecture does not hold for HE_7^(1) .
This is a joint work with Kailash Misra.