Periodic Reporting for period 2 - PosLieRep (Positivity in Lie groups and representation varieties)
Berichtszeitraum: 2022-11-01 bis 2024-04-30
Lie groups lie at the heart of mathematics. They play an important role in geometry, analysis, number theory, algebraic geometry and representation theory. As they describe symmetries of a space or a system they also appear prominently in theoretical physics. Not every system realizes the full amount of symmetry, it is therefore of key importance to investigate not only Lie groups, but also their subgroups, and in particular their discrete subgroups, which are often linked to geometric or arithmetic structures.
This projects builds upon new developments in the theory of Lie groups, in particular the introduction of total positivity in split real Lie groups on the one hand and new exiciting phenomena in the study of discrete subgroups, in particular the emergence of higher Teichmüller spaces.
Higher Teichmu ̈ller spaces generalize the classical theory of Fricke-Teichmüller space in the context of simple Lie groups of higher rank. The existence of higher Teichmüller spaces came as a surprise, and their discovery and investigation led to various other interesting developments, including an exciting interplay with the theory of Higgs bundles as well as with supersymmetric field theories in theoretical physics.
In this proposal we develop a unifying framework for higher Teichmüller spaces, which comprises the two known families, Hitchin components and maximal representations, but conjecturally also two new families. The basis for this conjectural unified theory lies in a new notion of positivity in Lie groups, which generalizes Lusztig’s total positivity in the context of arbitrary real Lie groups that are not necessarily split. This generalization of total positivity is of interest in its own right and leads to many exciting questions and conjectures that will be addressed in this proposal. The three main themes of the proposed project are Positivity in Lie groups, Positive representations as higher Teichmüller spaces, and Symplectic geometry of representation varieties.
This projects builds upon new developments in the theory of Lie groups, in particular the introduction of total positivity in split real Lie groups on the one hand and new exiciting phenomena in the study of discrete subgroups, in particular the emergence of higher Teichmüller spaces.
Higher Teichmu ̈ller spaces generalize the classical theory of Fricke-Teichmüller space in the context of simple Lie groups of higher rank. The existence of higher Teichmüller spaces came as a surprise, and their discovery and investigation led to various other interesting developments, including an exciting interplay with the theory of Higgs bundles as well as with supersymmetric field theories in theoretical physics.
In this proposal we develop a unifying framework for higher Teichmüller spaces, which comprises the two known families, Hitchin components and maximal representations, but conjecturally also two new families. The basis for this conjectural unified theory lies in a new notion of positivity in Lie groups, which generalizes Lusztig’s total positivity in the context of arbitrary real Lie groups that are not necessarily split. This generalization of total positivity is of interest in its own right and leads to many exciting questions and conjectures that will be addressed in this proposal. The three main themes of the proposed project are Positivity in Lie groups, Positive representations as higher Teichmüller spaces, and Symplectic geometry of representation varieties.
We introduced Theta-positivity, classified the semisimple Lie groups that admit a Theta positive structure, and proved that Theta positive representations give rise to higher Teichmüller spaces.
We worked on groups of type Sp_2 over non-commutative rings, proving generalizations of trace identities and corresponding quantum groups.
We also introduced generalizations of Thurston's stretch metric to spaces of Anosov representations, in particular to the Hitchin component.
We worked on groups of type Sp_2 over non-commutative rings, proving generalizations of trace identities and corresponding quantum groups.
We also introduced generalizations of Thurston's stretch metric to spaces of Anosov representations, in particular to the Hitchin component.
We expect to develop non-commutative coordinates on representation varieties and exhibit the corresponding non-commutative cluster algebras.
We will also work on establishing synthetic description of the Theta-positive semigroups, as well as possible non-negative semigroups.
We will also work on establishing synthetic description of the Theta-positive semigroups, as well as possible non-negative semigroups.