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Character (mathematics)
From Wikipedia, the free encyclopedia
There are several meanings of the word character in mathematics, although all are related to the idea of using fields (most of the time the complex numbers), to study a more abstract algebraic structure.
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If G is a group, a character is a group homomorphism into the multiplicative group of a field (as defined in Emil Artin's book on Galois Theory), usually the field of complex numbers. If A is an abelian group, then the set Ch(A) of these morphisms forms a group under the operation
- χaχb=χab.
This group is referred to as the character group. Sometimes only unitary characters are considered (so that the image is in the unit circle); other such homomorphisms are then called quasi-characters. Dirichlet characters can be seen a special case of this definition.
If f is a finite-dimensional representation of a group G, then the character of the representation is the function from G to the complex numbers given by the trace of f. In general, the trace is neither a group homomorphism, nor does the set of traces form a group. The study of representations by means of their characters is called character theory.
If A is an abelian algebra over the complex numbers, a character of A is an algebra homomorphism into the complex numbers. If in addition, A is a *-algebra, then a character is a *-homomorphism into the complex numbers.