  In this paper, we study the Green ring and the stable Green ring of a Hopf
algebra $H$ by means of bilinear forms. We show that the Green ring of a Hopf
algebra of finite representation type is a Frobenius algebra over $\mathbb{Z}$
with a dual basis associated to almost split sequences. On the stable Green
ring we define a new bilinear form which is more accurate to determine the
bi-Frobenius property of the stable Green ring. We show that the complexified
stable Green ring (or algebra) is a group-like algebra, and hence a
bi-Frobenius algebra, if the bilinear form on the stable Green ring is
non-degenerate.
