On the Grothendieck Ring of a Hopf Algebra

• The Grothendieck ring G0(H) is the abelian group generated by the iso. classes [V ] of finite dim’l (right) H-modules V modulo the relations [V ] = [U ] + [W ] for each s.e.s. 0 → U → V → W → 0. The iso. classes of simple modules form a Z-basis. Multiplication is given by ⊗ = ⊗k. • The character algebra R(H) is the k-subalgebra of H∗ that is generated by the image of the character map χ : G0(H) → H∗, [V ] 7→ χV , where χV (h) = traceV/k(v 7→ hv) .