, where the superscript is matrix transpose. Consistency requires With the definition given, For the Lie algebra representation one chooses consistency with a possible group representation. Generally, if is a representation of a Lie group, then given by is a representation of its Lie algebra. If is dual to , then its corresponding Lie algebra representation is given by Example Consider the group of complex numbers of absolute value 1. The irreducible representations are all one dimensional, as a consequence of Schur's lemma. The irreducible representations are parameterized by integers and given explicitly as The dual representation to is then the inverse of the transpose of this one-by-one matrix, that is, That is to say, the dual of the representation is . Generalization A general ring module does not admit a dual representation. Modules of Hopf algebras do, however. See also Complex conjugate representation Tensor product of representations Kirillov Character Formula References . Representation theory of groups