Standard Representations of Some Affine Lie Algebras

Abstract

The theory of Kac-Moody Lie algebras was originally developed by Kac [5] and Moody [19]. These are Lie algebras L = L(A) associated with generalized Cartan matrices (GCM) A — i.e. A = (aij) is an n × n integral matrix satisfying: (1) aii = 2, for ail i, (2) aij ⩽ 0, for all i ≠ j, and (3) aij = 0 ⇔ aji = 0 for all i ≠ j. The algebra L(A) is not necessarily finite-dimensional. Two GCM’s A = (aij) and B = (bij) are called equivalent if there is a permutation π of the indices such that bij = aπi, πj for all i,j. A GCM is indecomposable if it is not equivalent to a matrix in block form \( \left( {\begin{array}{*{20}{c}} * & 0 0 & * \end{array} } \right) \). The GCM A is symmetrizable if there exists a nonsingular diagonal matrix D such that DA is symmetric. The GCM A is said to be of finite type if it is the Cartan matrix of a finite-dimensional split semisimple Lie algebra. The GCM A is Euclidean if it is indecomposable, symmetrizable, singular and every principal submatrix is of finite type. The infinite-dimensional Kac-Moody Lie algebras associated with the Euclidean generalized Cartan matrices are called affine Lie algebras. These algebras are completely classified [5,20].