Free noncommutative principal divisors and commutativity of the tracial fundamental group
J. E. Pascoe

TL;DR
This paper introduces a theory of principal divisors for free noncommutative functions, compares their singularity sets, and explores the structure of fundamental groups, revealing they are abelian and composed of rational numbers.
Contribution
It defines principal divisors for free noncommutative functions and develops a cohomology and fundamental group theory for tracial free functions, highlighting their abelian nature.
Findings
Divisors of noncommutative rational functions are differences of polynomial divisors.
The fundamental group for tracial free functions is a direct sum of copies of .
Contrasts classical and free cases, showing abelian and rational structure.
Abstract
We define the principal divisor of a free noncommuatative function. We use these divisors to compare the determinantal singularity sets of free noncommutative functions. We show that the divisor of a noncommutative rational function is the difference of two polynomial divisors. We formulate a nontrivial theory of cohomology, fundamental groups and covering spaces for tracial free functions. We show that the natural fundamental group arising from analytic continuation for tracial free functions is a direct sum of copies of . Our results contrast the classical case, where the analogous groups may not be abelian, and the free case, where free universal monodromy implies such notions would be trivial.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Algebra and Geometry · Geometric and Algebraic Topology
