Positivity and Non-commutativity

TL;DR

This paper explores a generalized notion of positivity in real semisimple Lie groups, connecting it to higher rank Teichmüller spaces and non-commutative structures, with applications like non-commutative Markov numbers.

Contribution

It introduces a unified framework for positivity in various Lie groups and proposes a non-commutative perspective with novel applications.

Findings

Generalizes total positivity to broader Lie groups

Links positivity with higher rank Teichmüller spaces

Suggests non-commutative analogs of Markov numbers

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 Teichm\"uller 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.