Product Information
Description
An annual survey of the math.QA arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.
Book framework (share of the year's output):
- Quantum Groups and Discrete Quantum Symmetries (17.2%) — Investigates structural properties, classification, and representation theory of quantum groups, discrete quantum groups, and related algebraic symmetries.
- Affine Algebras, Yangians, and Twisted Structures (17.1%) — Focuses on representations, deformations, and integrable structures of affine Lie algebras, Yangians, and their twisted variants.
- Vertex Operator Algebras and Conformal Field Theory (13.4%) — Explores module categories, character identities, and modular invariance within vertex operator algebras and their connections to conformal field theory.
- Tensor Categories and Modular Fusion Systems (10.2%) — Analyzes the classification and functorial properties of fusion categories, modular tensor categories, and their applications in topological quantum field theories.
- Hopf Algebras and Bialgebraic Deformations (9.2%) — Studies extensions, deformations, and classification of Hopf algebras, including pointed, cocommutative, and weak Hopf algebra structures.
- Lie Algebras, W-Algebras, and Hecke Structures (7.5%) — Examines representations and integrable systems associated with Lie algebras, W-algebras, Hecke algebras, and their Poisson or cluster deformations.
- Topological Quantum Field Theories and Quantum Graphs (6.9%) — Connects quantum cellular automata, quantum graphs, and topological quantum field theories through algebraic structures like Frobenius subalgebras and cluster systems.
- Noncommutative Geometry and Topological Invariants (6.8%) — Applies noncommutative geometric methods and homological algebra to construct topological invariants for manifolds, links, and skein modules.
- Quantum Integrable Systems and Special Functions (6.0%) — Develops algebraic frameworks for quantum integrable models, special functions, and their relations to quantum groups and representation theory.
- Quantum Algebraic Structures and Langlands Correspondence (5.7%) — Explores advanced algebraic structures including braided automata, skew braces, and two-parameter Langlands correspondences within quantum algebra.
Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.
