Lattice endomorphisms, Seifert forms and upper triangular matrices

Larabi, Khadija

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URN:	urn:nbn:de:bsz:180-madoc-695818
Document Type:	Doctoral dissertation
Year of publication:	2025
Place of publication:	Mannheim
University:	Universität Mannheim
Evaluator:	Hertling, Claus
Date of oral examination:	2025
Publication language:	English
Institution:	School of Business Informatics and Mathematics > Algebraische Geometrie (Hertling 2005-)
Subject:	510 Mathematics
Keywords (English):	unimodular bilinear lattice , upper triangular matrix , Seifert form , even and odd intersection form , monodromy group , vanishing cycle , braid group action , distinguished basis
Abstract:	This monograph starts with an upper triangular matrix with integer entries and 1’s on the diagonal. It develops from this a spectrum of structures, which appear in different contexts, in algebraic geometry, representation theory and the theory of irregular meromorphic connections. It provides general tools to study these structures, and it studies sytematically the cases of rank 2 and 3. The rank 3 cases lead already to a rich variety of phenomena and give an idea of the general landscape. Their study takes up a large part of the monograph. Special cases are related to Coxeter groups, generalized Cartan lattices and exceptional sequences, or to isolated hypersurface singularities, their Milnor lattices and their distinguished bases. But these make only a small part of all cases. One case in rank 3 which is beyond them, is related to quantum cohomology of P2 and to Markov triples. The first structure associated to the matrix is a Z-lattice with a unimodular bilinear form (called Seifert form) and a triangular basis. It leads immediately to an even and an odd intersection form, reflections and transvections, an even and an odd monodromy group, even and odd vanishing cycles. Braid group actions lead to braid group orbits of distinguished bases and of upper triangular matrices.