Positivity and Non-commutativity

2025/11/16 by Anna Wienhard, Wienhard, Anna
Mathematics · #22E40 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometry and complex manifolds #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2511.12771

openalex publication_date 2025/11/16 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28

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.

Citations

• Noncommutative Polygonal Cluster Algebras
• Theta Positivity in Lagrangian Grassmannian
• On partial abelianization of framed local systems
• A shadow Markov equation
• Spectral description of non-commutative local systems on surfaces and non-commutative cluster varieties
• Positivity and representations of surface groups
• Positive surface group representations in PO(p,q)
• Symplectic groups over noncommutative algebras
• Noncommutative coordinates for symplectic representations
• Quantum geometry of moduli spaces of local systems and representation theory
• Geometric structures and representations of discrete groups
• Positivity and higher Teichmüller theory
• Noncommutative marked surfaces
• Canonical bases for cluster algebras
• Dual Teichmuller and lamination spaces
• Moduli spaces of local systems and higher Teichmuller theory
• Surface group representations with maximal Toledo invariant
• An algebraic cell decomposition of the nonnegative part of a flag variety

Related

• Positivity and higher Teichmüller theory
• Proceedings of the Young Researchers Workshop on Positivity in Lie Groups
• Generalizing Lusztig's total positivity
• A survey of total positivity
• From Weyl groups to semisimple groups