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A proper generating functional on a Podleś sphere

Masato Tanaka

Abstract

We construct a proper generating functional $L$ on a Podleś sphere and we show that $1$-cocycle arising from $L$ coincides with the one in our previous work. We also show that our 1-cocycle is purely non Gaussian and that the full `group' $C^{\ast}$-algebra of the quantum $SL(2,\mathbb{R})$ is liminal.

A proper generating functional on a Podleś sphere

Abstract

We construct a proper generating functional on a Podleś sphere and we show that -cocycle arising from coincides with the one in our previous work. We also show that our 1-cocycle is purely non Gaussian and that the full `group' -algebra of the quantum is liminal.
Paper Structure (10 sections, 17 theorems, 47 equations)

This paper contains 10 sections, 17 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Observations of conventions
  5. The algebras $\mathcal{U}_q$ and $U_q(\mathfrak{su}(2))$
  6. Left v.s. right
  7. Observations of representations
  8. Generating functionals and $1$-cocycles
  9. Some more observations
  10. Research directions

Key Result

Lemma 3.2

(cf. K) The element $u_{[a],[a]}^n$ is a polynomial in $u_{[a],[a]}^1$ of degree $n$.

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 3.1
  • Lemma 3.2
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Proposition 4.3
  • ...and 29 more