Algebra representation of a Hopf algebra - Definition and Overview

You know what an ordinary (vector space) representation of a Hopf algebra is. It turns out Hopf algebras also have algebra reps with an additional structure over and above the module structure.

Let H be a Hopf algebra. If A is an algebra with the product operation <math>\mu:A\otimes A\rightarrow A<math>, then a linear map <math>\rho:H\otimes A\rightarrow A<math> is an algebra representation of H if, in addition to being a (vector space) rep of H, <math>\mu<math> is an H-intertwiner. Recall that <math>A\otimes A<math> is also a vector space rep of H. If A happens to be unital, we'd require that there's an H-intertwiner from ε_H to A such that the 1 of ε_H maps to the unit of A.

Algebra representation of a Lie algebra, Algebra representation of a Lie superalgebra and Algebra representation of a group are all special cases of this more general concept.