Triangular Bases of Unimodular Bilinear Lattices and Induced Structures
Description
This monograph presents a structured and conceptually unified study of upper triangular integer matrices and the rich family of structures they induce. Central objects include unimodular bilinear lattices, vanishing cycles, monodromy groups, braid group actions, and associated moduli spaces. These constructions naturally provide a common framework linking algebraic geometry, representation theory, singularity theory, and the theory of irregular meromorphic connections.
To a given matrix is associated a Z-lattice with a unimodular bilinear form (the Seifert form) and a triangular basis. This leads to even and odd intersection forms, reflections and transvections, monodromy groups, and corresponding vanishing cycles. Braid group actions generate orbits of distinguished bases and matrices, which in turn give rise to complex manifolds obtained by gluing Stokes regions.
General tools and results are developed throughout, with a systematic analysis of the cases of rank 2 and 3. Already in rank 3 a wide range of phenomena appears, illustrating the broader landscape. Classical situations related to Coxeter groups, generalized Cartan lattices, exceptional sequences, and isolated hypersurface singularities arise as special cases but represent only a small part of the theory. The book is intended for researchers and students working with integral upper triangular matrices and their induced structures.
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Persons
Claus Hertling is a Professor at the University of Mannheim. His research interests are in isolated hypersurface singularities and their moduli spaces, Frobenius manifolds, meromorphic connections and, more recently, integral matrices in the contexts of Stokes structures and monodromy.
Khadija Larabi obtained her PhD from the University of Mannheim. She works on algebraic structures induced by upper triangular integral matrices and on orders and full lattices in finite-dimensional commutative Q-algebras.
Content
Chapter 1. Introduction.- Chapter 2. Bilinear lattices and induced structures.- Chapter 3. Braid group actions.- Chapter 4. Braid group action on upper triangular 3×3 matrices.- Chapter 5. Automorphism groups.-Chapter 6. Monodromy groups and vanishing cycles.- Chapter 7. Distinguished bases in the rank 2 and rank 3 cases.- Chapter 8. Manifolds induced by the orbit of distinguished bases
and the orbit of distinguished matrices.- Chapter 9. The manifolds in the rank 3 cases.- Chapter 10. Isolated hypersurface singularities.- Appendix A. Tools from hyperbolic geometry.- Appendix B. The first congruence subgroups.- Appendix C. Quadratic units via continued fractions.- Appendix D. Powers of quadratic units.- Bibliography.- Index.