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

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.

20 pages, correction of a wrong argument in the proof of Theorem 5.1 (result unchanged). Final version, to appear in Osaka J. Math

References in corpus (2)

• Radford, Drinfeld, and Cardy boundary states in (1,p) logarithmic conformal field models
• Logarithmic Hennings invariants for restricted quantum sl(2)

Cited by in corpus (1)

• Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras