Triangular Bases of Unimodular Bilinear Lattices and Induced Structures: Automorphism Groups, Vanishing Cycles, Monodromy Groups, Braid Group Actions
暫譯: 單模雙線性格的三角基底與誘導結構：自同構群、消滅循環、單值群與辮群作用

Hertling, Claus, Larabi, Khadija

Triangular Bases of Unimodular Bilinear Lattices and Induced Structures: Automorphism Groups, Vanishing Cycles, Monodromy Groups, Braid Group Actions
  • 出版商: Springer
  • 出版日期: 2026-09-23
  • 售價: $3,680
  • 貴賓價: 9.5 折 $3,496
  • 語言: 英文
  • 頁數: 382
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3032259967
  • ISBN-13: 9783032259967
  • 相關分類: 數學

    • 海外代購書籍(需單獨結帳)

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商品描述

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.

商品描述(中文翻譯)

本專論對上三角整數矩陣及其所誘導的豐富結構，提出了具系統性且概念統一的研究。核心對象包括單模雙線性格（unimodular bilinear lattices）、消失循環（vanishing cycles）、單值群（monodromy groups）、編織群作用（braid group actions），以及相關的模空間（moduli spaces）。這些構造自然地提供了一個共同框架，將代數幾何、表示論、奇點理論，以及不規則亞純連接（irregular meromorphic connections）理論連結起來。

對於給定的矩陣，可對應到一個帶有單模雙線性形式（Seifert 形式）與三角形基底的 ℤ-格。由此可導出偶交形式與奇交形式、反射與橫截變換（transvections）、單值群，以及相應的消失循環。編織群作用會生成特殊基底與矩陣的軌道，而這些軌道進一步產生透過黏合 Stokes 區域所得到的複流形。

全書逐步發展一般性的工具與結果，並系統性地分析秩為 2 與 3 的情形。即使在秩 3 的情況下，也已經出現廣泛多樣的現象，展現出更宏觀的理論面貌。與 Coxeter 群、廣義 Cartan 格、例外序列（exceptional sequences）及孤立超曲面奇點（isolated hypersurface singularities）相關的經典情形，都是本理論的特殊案例，但僅構成其中一小部分。本書適合研究整數上三角矩陣及其所誘導結構的研究人員與學生閱讀。

作者簡介

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.

作者簡介(中文翻譯)

Claus Hertling 是 Mannheim 大學教授。他的研究興趣包括孤立超曲面奇點及其模空間、Frobenius 流形、亞純連接，以及近年來 Stokes 結構與單值化背景下的整數矩陣。

Khadija Larabi 取得 Mannheim 大學博士學位。她的研究領域包括由上三角整數矩陣所誘導的代數結構，以及有限維交換 ℚ-代數中的階（orders）與滿格（full lattices）。

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