A characterization of generalized Kac - Moody algebras

Generalized Kac-Moody algebras can be described in two ways: either using generators and relations, or as Lie algebras with an almost positive definite symmetric contravariant bilinear form. Unfortunately it is usually hard to check either of these conditions for any naturally occurring Lie algebra. In this paper we give a third characterization of generalized Kac-Moody algebras which is easier to check, which says roughly that any Lie algebra with a root system similar to that of a generalized Kac-Moody algebra is a generalized Kac-Moody algebra. We use this to show that some Lie algebras constructed from even Lorentzian lattices are generalized Kac-Moody algebras. I thank the referee for suggesting several improvements and corrections. Section 1 states the theorems of this paper, section 2 describes some examples, and section 3 gives the proof of the theorems. 1. Statement of result. All Lie algebras are Lie algebras over the real numbers R. We will assume the basic theory of generalized Kac-Moody algebras given in [2,3,4]. We first recall the definition of a generalized Kac-Moody algebra. Suppose that aij is a real square matrix indexed by i and j in some countable set I with the following properties. 1 aij = aji. 2 If i 6= j then aij ≤ 0. 3 If aii > 0 then 2aij/aii is an integer for all j. Then we define the universal generalized Kac-Moody algebra of aij to be the Lie algebra generated by elements ei, fi, and hij for i, j ∈ I, with the following relations. 1 [ei, fj ] = hij . 2 [hij , ek] = δ j i aikek, [hij , fk] = −δ j i aikfk. 3 If aii > 0 then Ad(ei)ijiiej = Ad(fi)ijiifj = 0. 4 If aij = 0 then [ei, ej ] = [fi, fj ] = 0. (The relations [hij , hkl] are usually also included, but these follow from the other relations.) We define a generalized Kac-Moody algebra to be a Lie algebra G such that G is a semidirect product A.B, where A is an ideal of G which is the quotient of a universal generalized Kac-Moody algebra by a subspace of its center, and B is an abelian subalgebra such that the elements ei and fi are all eigenvalues of B. In other words, G can be obtained from a universal generalized Kac-Moody algebra by throwing away some of the center and adding some commuting outer derivations. This is slightly more restrictive than some previous definitions because we insist that I is a countable set, but I do not know of any interesting examples where I is uncountable. Kac [4] extended the definition by allowing the matrix aij to be non symmetrizable. In this case the generalized Kac-Moody algebra does not have an invariant bilinear form, and the existence of such a form is an essential condition in the theorems of this paper. I do not