In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known example is the monster Lie algebra. Motivation Finite-dimensional semisimple Lie algebras have the following properties: They have a nondegenerate symmetric invariant bilinear form (,). They have a grading such that the degree zero piece (the Cartan subalgebra) is abelian. They have a (Cartan) involution w. (a, w(a)) is positive if a is nonzero. For example, for the algebras of n by n matrices of trace zero, the bilinear form is (a, b) = Trace(ab), the Cartan involution is given by minus the transpose, and the grading can be given by "distance from the diagonal" so that the Cartan subalgebra is the diagonal elements. Conversely one can try to find all Lie algebras with these properties (and satisfying a few other technical conditions). The answer is that one gets sums of finite-dimensional and affine Lie algebras. The monster Lie algebra satisfies a slightly weaker version of the conditions above: (a, w(a)) is positive if a is nonzero and has nonzero degree, but may be negative when a has degree zero. The Lie algebras satisfying these weaker conditions are more or less generalized Kac–Moody algebras. They are essentially the same as algebras given by certain generators and relations (described below). Informally, generalized Kac–Moody algebras are the Lie algebras that behave like finite-dimensional semisimple Lie algebras. In particular they have a Weyl group, Weyl character formula, Cartan subalgebra, roots, weights, and so on. Definition A symmetrized Cartan matrix is a (possibly infinite) square matrix with entries such that if is an integer if The universal generalized Kac–Moody algebra with given symmetrized Cartan matrix is defined by generators and and and relations if , 0 otherwise , for applications of or if if These differ from the relations of a (symmetrizable) Kac–Moody algebra mainly by allowing the diagonal entries of the Cartan matrix to be non-positive. In other words, we allow simple roots to be imaginary, whereas in a Kac–Moody algebra simple roots are always real. A generalized Kac–Moody algebra is obtained from a universal one by changing the Cartan matrix, by the operations of killing something in the center, or taking a central extension, or adding outer