A note on symmetric linear forms and traces on the restricted quantum group $\bar U_q(\mathfrak{sl}(2))$

Abstract

In this paper we prove two results about ${\rm SLF}(\bar U_q)$, the algebra of symmetric linear forms on the restricted quantum group $\bar U_q = \bar U_q\left(\mathfrak{sl}(2)\right)$. First, we express any trace on finite dimensional projective $\bar U_q$-modules as a linear combination in the basis of ${\rm SLF}(\bar U_q)$ constructed by Gainutdinov - Tipunin and also by Arike. In particular, this allows us to determine the symmetric linear form corresponding to the modified trace on projective $\bar U_q$-modules. Second, we give the explicit multiplication rules between symmetric linear forms in this basis.