  We study Hopf algebras via tools from geometric invariant theory. We show
that all the invariants we get can be constructed using the integrals of the
Hopf algebra and its dual together with the multiplication and the
comultiplication, and that these invariants determine the isomorphism class of
the Hopf algebra. We then define certain canonical subspaces $Inv^{i,j}$ of
tensor powers of $H$ and $H^*$, and use the invariant theory to prove that
these subspaces satisfy a certain non-degeneracy condition. Using this
non-degeneracy condition together with results on symmetric monoidal
categories, we prove that the spaces $Inv^{i,j}$ can also be described as
$(H^{\otimes i}\otimes (H^*)^{\otimes j})^A$, where $A$ is the group of Hopf
automorphisms of $H$. As a result we prove that the number of possible Hopf
orders of any semisimple Hopf algebra over a given number ring is finite. we
give some examples of these invariants arising from the theory of
Frobenius-Schur Indicators, and from Reshetikhin-Turaev invariants of three
manifolds. We give a complete description of the invariants for a group
algebra, proving that they all encode the number of homomorphisms from some
finitely presented group to the group. We also show that if all the invariants
are algebraic integers, then the Hopf algebra satisfies Kaplansky's sixth
conjecture: the dimensions of the irreducible representations of $H$ divide the
dimension of $H$.
