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Positivity and Non-commutativity

Anna Wienhard

Abstract

In this article we revisit a new notion of positivity in real semisimple Lie groups that at the same time generalizes total positivity in split real Lie groups as well as positive Lie semigroups in Hermitian Lie groups of tube type. We shortly discuss the relationship with higher rank Teichmüller spaces, and then focus on describing different aspects of positivity as well as open questions. In the second part we describe a non-commutative perspective on Hermitian Lie groups of tube type that is suggested by positivity and leads to interesting applications, such as non-commutative generalizations of Markov numbers.

Positivity and Non-commutativity

Abstract

In this article we revisit a new notion of positivity in real semisimple Lie groups that at the same time generalizes total positivity in split real Lie groups as well as positive Lie semigroups in Hermitian Lie groups of tube type. We shortly discuss the relationship with higher rank Teichmüller spaces, and then focus on describing different aspects of positivity as well as open questions. In the second part we describe a non-commutative perspective on Hermitian Lie groups of tube type that is suggested by positivity and leads to interesting applications, such as non-commutative generalizations of Markov numbers.

Paper Structure

This paper contains 25 sections, 3 theorems, 49 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Positivity in Lie groups
  3. The four families
  4. A non-commutative view on the special piece $\mathfrak{u}_{\alpha_\Theta}$
  5. The unipotent semigroup
  6. The $\Theta$-Weyl group
  7. Parametrizing the positive semigroup
  8. Parametrizations of the non-negative semigroups
  9. Positive parts of flag varieties
  10. Aspects of $\mathrm{SL}_2(\mathbb{R})$
  11. $\mathrm{SL}_2(\mathbb{R})$ as $\mathrm{Sp}_2(\mathbb{R})$
  12. The symmetric space
  13. Trace Identities
  14. Markov numbers
  15. Cluster algebras
  16. ...and 10 more sections

Key Result

Theorem 2.1

GW_pos1 A simple Lie group $G$ admits a positive structure with respect to $\Theta$ if and only if the pair $(G,\Theta)$ belongs to the following list:

Figures (5)

  • Figure 1: Mutation at the arc $\gamma$.
  • Figure 2: The Markov quiver and a seed associated to a triangulation of a punctured torus
  • Figure 3: An ideal triangulation of a punctured torus with a horocycle. Picture taken from Teschner.
  • Figure 4: Seed associated to a triangulation of a punctured torus
  • Figure 5: The Markov "tree" for complex Markov numbers

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 4.1
  • Conjecture 8.1