Editing Dimension (vector space) (section)

=== Trace ===
{{see also|Trace (linear algebra)}}

The dimension of a vector space may alternatively be characterized as the [[Trace (linear algebra)|trace]] of the [[identity operator]]. For instance, 
<math>\operatorname{tr}\ \operatorname{id}_{\R^2} = \operatorname{tr} \left(\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right) = 1 + 1 = 2.</math> 
This appears to be a [[circular definition]], but it allows useful generalizations.

Firstly, it allows for a definition of a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an [[Algebra over a field|algebra]] <math>A</math> with maps <math>\eta : K \to A</math> (the inclusion of scalars, called the ''unit'') and a map <math>\epsilon : A \to K</math> (corresponding to trace, called the ''[[counit]]''). The composition <math>\epsilon \circ \eta : K \to K</math> is a scalar (being a linear operator on a 1-dimensional space) corresponds to "trace of identity", and gives a notion of dimension for an abstract algebra. In practice, in [[bialgebra]]s, this map is required to be the identity, which can be obtained by normalizing the counit by dividing by dimension (<math>\epsilon := \textstyle{\frac{1}{n}} \operatorname{tr}</math>), so in these cases the normalizing constant corresponds to dimension.

Alternatively, it may be possible to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of "[[trace class]] operators" on a [[Hilbert space]], or more generally [[nuclear operator]]s on a [[Banach space]].

A subtler generalization is to consider the trace of a ''family'' of operators as a kind of "twisted" dimension. This occurs significantly in [[representation theory]], where the [[Character (mathematics)|character]] of a representation is the trace of the representation, hence a scalar-valued function on a [[Group (mathematics)|group]] <math>\chi : G \to K,</math> whose value on the identity <math>1 \in G</math> is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: <math>\chi(1_G) = \operatorname{tr}\ I_V = \dim V.</math> The other values <math>\chi(g)</math> of the character can be viewed as "twisted" dimensions, and find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of [[monstrous moonshine]]: the [[j-invariant|<math>j</math>-invariant]] is the [[graded dimension]] of an infinite-dimensional graded representation of the [[monster group]], and replacing the dimension with the character gives the [[McKay–Thompson series]] for each element of the Monster group.<ref name="gannon">{{Citation|last=Gannon|first=Terry|title=Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics|year=2006|publisher=Cambridge University Press |isbn=0-521-83531-3}}</ref>