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

Matthieu Faitg

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.

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 , the algebra of symmetric linear forms on the restricted quantum group . First, we express any trace on finite dimensional projective -modules as a linear combination in the basis of 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 -modules. Second, we give the explicit multiplication rules between symmetric linear forms in this basis.

Paper Structure

This paper contains 11 sections, 12 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The restricted quantum group $\bar{U}_q(\mathfrak{sl}(2))$
  4. Simple and projective $\bar{U}_q$-modules
  5. Structure of the bimodule $_{\bar{U}_q}\!\left(\bar{U}_q\right)_{\bar{U}_q}$ and the center of $\bar{U}_q$
  6. Symmetric linear forms and the GTA basis
  7. Traces on projective $\bar{U}_q$-modules and the GTA basis
  8. Correspondence between traces and symmetric linear forms
  9. Link with the GTA basis
  10. Symmetric linear form corresponding to the modified trace on $\mathop{\mathrm{Proj}}\nolimits_{\bar{U}_q}$
  11. Multiplication rules in the GTA basis

Key Result

Lemma 2.1

The right $\bar{U}_q$-module $R^*(\mathcal{X}^{\alpha}(s))$ admits a basis $\left(\bar{v}_i\right)_{0 \leq i \leq s-1}$ such that The right $\bar{U}_q$-module $R^*(\mathcal{P}^{\alpha}(s))$ admits a basis $\left(\bar{b}_{i}, \bar{x}_j, \bar{y}_k, \bar{a}_l\right)_{\substack{0 \leq i,l \leq s-1 \\ 0 \leq j ,k\leq p-s-1}}$ such that

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • Definition 3.1
  • Remark 1
  • ...and 16 more