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

TL;DR

This paper characterizes symmetric linear forms on the restricted quantum group ar U_q(sl(2)) by expressing traces on projective modules in a specific basis and providing explicit multiplication rules, advancing understanding of quantum algebra structures.

Contribution

It explicitly describes the basis of symmetric linear forms and their multiplication, linking traces on projective modules to this basis in the context of ar U_q(sl(2)).

Findings

Expressed any trace on projective modules as a linear combination in a specific basis.

Determined the symmetric linear form corresponding to the modified trace.

Provided explicit multiplication rules between symmetric linear forms.

Abstract

We prove two results about $\text{SLF}(\bar U_q)$, the algebra of symmetric linear forms on the restricted quantum group $\bar U_q = \bar U_q(\mathfrak{sl}(2))$. First, we express any trace on finite dimensional projective $\bar U_q$-modules as a linear combination in the basis of $\text{SLF}(\bar U_q)$ 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.