Lie-like Approach to the Theory of Representations of Finite Groups

Abstract

Using characters, one can set up the theory of representations for both finite and continuous groups. For continuous groups, another approach—the Lie theory—is also possible. It is shown that a similar theory, based on commutators, can be developed also in the case of finite groups. Essentially, the group algebra of a finite group is converted into a Lie algebra by replacing the usual associative product by the product x∘y=xy−yx. Then the resulting Lie algebra is a direct sum of special unitary Lie algebras.