Hopf algebra polynomial identities in the Sweedler algebra

Abstract

Using Mac Lane's PROPs, one can introduce multilinear polynomial identities that involve all the operations of a Hopf algebra: the (co)multiplication, the (co)unit and the antipode. We find bases for polynomial identities and calculate their codimensions for the Sweedler algebra, treating it first as an ordinary associative algebra over a field and second as a Hopf algebra.