KAC-MOODY ALGEBRAS WITH NONSYMMETRIZABLE CARTAN MATRICES (NONASSOCIATIVE ALGEBRA, LIE THEORY, SYMMETRIZATION)

The Kac-Moody algebra L = L(A) = (SIGMA) L(,(alpha)) = L/M is the quotient of the maximal graded Lie algebra L having A as its matrix of fundamental structure constants (Cartan matrix) by the homogeneous ideal M maximal with respect to trivially intersecting the 0-grade space (Cartan subalgebra) h of L. Robert Moody and Victor Kac independently introduced these algebras and proved detailed structural properties for L/center(L) when their (generalized) Cartan matrices A are symmetrizable. This dissertation presents structural results for L when the Cartan matrix is not symmetrizable.;It is known, when the Cartan matrix A is combinatorially symmetric, that M contains the ideal S generated by the Serre relations. Our investigation of S proves that it nontrivially intersects h when the Cartan matrix is combinatorially asymmetric. Indeed, sometimes S = L. Surprisingly, however, we can produce nontrivial invariant bilinear forms for some Kac-Moody algebras having indecomposable, nonsymmetrizable Cartan matrices.;We begin with a brief study of symmetrizability, proving that a matrix is not symmetrizable if and only if either the matrix is not combinatorially symmetric or the matrix has a nonsymmetrizable submatrix whose graph is a simple cycle (and the product of the cycle is not equal to its transpose product). Turning our attention to L we prove that dim L(,(alpha)),L(,-(alpha)) > 1 for certain formal weights (alpha) when the product of some nonrepeating closed path of A is not equal to its transpose product, and we use the Weyl group to generalize this result slightly. In contrast, we prove that some Kac-Moody algebras having combinatorially asymmetric (hence nonsymmetrizable) 2 x 2 Cartan matrices have dim L(,(alpha)),L(,-(alpha)) (LESSTHEQ) 1 for all (alpha).