Editing Quaternionic representation (section)

{{Short description|Representation of a group or algebra in terms of an algebra with quaternionic structure}}
In the [[mathematics|mathematical]] field of [[representation theory]], a '''quaternionic representation''' is a [[group representation|representation]] on a [[complex number|complex]] vector space ''V'' with an invariant [[quaternionic structure]], i.e., an [[antilinear]] [[equivariant map]]

:<math>j\colon V\to V</math> 
which satisfies

:<math>j^2=-1.</math>

Together with the imaginary unit ''i'' and the antilinear map ''k''&nbsp;:=&nbsp;''ij'', ''j'' equips ''V'' with the structure of a [[quaternionic vector space]] (i.e., ''V'' becomes a [[module (mathematics)|module]] over the [[division algebra]] of [[quaternion]]s). From this point of view, quaternionic representation of a [[group (mathematics)|group]] ''G'' is a [[group homomorphism]] ''&phi;'': ''G'' &rarr; GL(''V'',&nbsp;'''H'''), the group of invertible quaternion-linear transformations of ''V''. In particular, a quaternionic matrix representation of ''g'' assigns a [[square matrix]] of quaternions ''&rho;''(g) to each element ''g'' of ''G'' such that ''&rho;''(e) is the [[identity matrix]] and

:<math>\rho(gh)=\rho(g)\rho(h)\text{ for all }g, h \in G.</math>

Quaternionic representations of [[associative algebra|associative]] and [[Lie algebra]]s can be defined in a similar way.