Fuzzy Properties on Finite Type of Kac-Moody Algebras

Abstract

The theory of Kac-Moody algebra was introduced by V.G. Kac & R.V. Moody simultaneously and independently in 1968. The subject Kac-Moody algebras attracted the attention of many Mathematicians because of its various connections and applications to different branches of Mathematics and Mathematical Physics. On the other hand, the concept of fuzzy sets originated in a seminar paper by Lotfi A. Zadeh in 1965. Since then, the subject has developed in leaps and bounds and finds its application in all branches of science, engineering, technology, economics, social and behavioral sciences. In this paper, we introduce the notion of fuzzy sets on Kac-Moody Lie algebras. We define a fuzzy set on the set of simple roots using the Generalized Cartan Matrix elements. Some basic properties like convexity, cardinality, normality are studied. We also compute the α – level sets and strong α – level sets in the case of finite type of Kac-Moody algebras D_l (l ≥ 4), E₆ , E₇,E₈ , F₄ and G₂.