94 08 03 7 v 1 5 A ug 1 99 4 Realizations of the Monster Lie Algebra

In this paper we reinterpret the theory of generalized Kac-Moody Lie algebras in terms of local Lie algebras formed from reductive or Kac-Moody algebras and certain modules (possibly infinite-dimensional) for these algebras. We exploit the fact, established in [19], that certain generalized Kac-Moody algebras contain specific large free subalgebras, in order to exhibit these generalized KacMoody algebras as explicitly prescribed Lie algebras of operators acting on tensor algebras. In the most important special case, that of R. Borcherds’ Monster Lie algebra m, introduced in [4] (see also [5]), we apply the free subalgebra result [19] to simplify Borcherds’ work [4] on the Conway-Norton conjectures (see [8]) for the “moonshine module” V ♮ ([11], [12]) for the Fischer-Griess Monster group M . In particular, we realize m as an explicitly prescribed M -covariant Lie algebra of operators acting on the tensor algebra over a certain gl2and M -module built in a simple way from the M -module V . In [4], Borcherds produces recursion relations which, along with initial conditions, uniquely characterize the McKay-Thompson series (i.e., the graded traces) of the elements of M acting on the infinite-dimensional M -module V ♮ constructed in [11], [12]. In this way he completes the proof of the Conway-Norton conjectures as they relate to V , in the sense that he shows that the McKayThompson series for V ♮ do indeed agree with the modular functions listed in [8] for all elements of M , since the coefficients of those modular functions satisfy the same recursion relations (replication formulas) and initial conditions. For instance, for the identity element of M the corresponding McKay-Thompson