There are many notions of a character for an algebraic structure, often topologized.
A character on the group is a homomorphism into the group of units of the ground field. Regarding that the codomain is abelian, the collection of characters is itself an abelian group under the pointwise multiplication, this is called the character lattice
of the group. Similarly the cocharacter lattice is .
For more see at group character.
For topological groups one considers continuous characters. Specifically, for a locally compact Hausdorff group (often further assumed to be an abelian group), a character of is continuous homomorphism to the circle group . If is profinite, then this is the same as an continuous homomorphism to the discrete group . (See MO.)
In representation theory, one defines the character of a representation to be the function on given by , whenever the trace in makes sense (e.g. when is finite-dimensional). Since such a function is invariant under conjugation, we may equivalently consider it a function on the set of conjugacy classes of elements in .
Sometimes we also extend a character linearly to the free vector space on the set of conjugacy classes. This version of the character can be identified with the bicategorical trace of the identity map of the representation, considered as a --module.
There is a different notion of an infinitesimal character in Harish–Chandra theory and also a notion of the formal character.
There are important formulas concerning characters in representation theory, like Weyl character formula, Kirillov character formula, Demazure character formula and so on. There is also a formula for the induced character of an induced representation.
Let be a unital, not necessarily commutative, ring (or, more generally, -algebra for -commutative); then a monoid in a category of -bimodules (which are respectively also compatibly -modules), is called an -ring. In other words an -ring is an object in the coslice category ; it is thus a ring equipped with multiplication and a map of rings.
A left character of an -ring is a map such that
(i) (left -linearity) for all ,
(ii) (associativity) for all
where we denoted multiplication in and in by concatenation. The conditions on can be restated as the requirement that the map given by is a -action extending the left regular -action (i.e. the multiplication on considered as a left action).
Dually, a right character of an -ring is a map such that
(i) (right -linearity) for all ,
(ii) (associativity) for all
This is in turn equivalent to extending the right regular action of to the action of on .
The character of a topological space at a point is the minimal cardinality of a local basis of neighborhoods of point (local basis of topology on ) if it is infinite and aleph zero otherwise. The character of a topological space is the supremum of when runs through .
Character rings of compact Lie groups are discussed in
Masaru Tackeuchi, A remark on the character ring of a compact Lie group, J. Math. Soc. Japan Volume 23, Number 4 (1971), 555-705 (Euclid)
Troels Roussauc Johansen, Character Theory for Finite Groups and Compact Lie Groups pdf