A Class of Lie Algebras That Contains a Class of Kac-Moody Algebras

V. Kac showed that from a local Lie algebra, we can construct a graded Lie algebra whose local part is the given local Lie algebra. Later, the author defined the notion of standard pentads. Using this concept, we can construct a local Lie algebra from a given Lie algebra and its representation. Therefore, by using standard pentads, we can construct a graded Lie algebra that has a local part consisting of a given Lie algebra and its representation under certain assumptions. As special cases of Lie algebras associated with standard pentads, we have the notion of PC Lie algebras. A PC Lie algebra is a graded Lie algebra that is constructed from a finite-dimensional commutative Lie algebra and its finite-dimensional completely reducible representation. Our aim in this paper is to show that the class of PC Lie algebras contains the class of Kac-Moody algebras.