A Characterization of Linearly Semisimple Groups

Let $G=SpecA$ be an affine $K$-group scheme and $\tilde{A} = {w \in A*: dim_K A^* \cdot w \cdot A^* < \infty }$\mathrm{.}Let$< -,-> : A^* \times \tilde{A} \to K, (w,\tilde{w}) := tr(w \tilde{w})$,bethetraceform\mathrm{.}Weprovethat$G$islinearlyreductiveifandonlyifthetraceformisnon-degenerateon$A^*$.