EXTENDING TABLE ALGEBRAS TO HOPF ALGEBRAS

Let $A$ be a table algebra with standard basis $\mathbf{B}$, multiplication $\mu$, unit map $\eta$, skew-linear involution $*$, and degree map $\delta$.  In this article we study the possible coalgebra structures $(A,\Delta, \delta)$ on $A$ for which $(A, \mu, \eta, \Delta, \delta)$ becomes a Hopf algebra with respect to some antipode.  We show that such Hopf algebra structures are not always available for noncommutative table algebras.  On the other hand, commutative table algebras will always have a Hopf algebra structure induced from an algebra-isomorphic group algebra.  To illustrate our approach, we derive Hopf algebra comultiplications on table algebras of dimension 2 and 3.