Braid groups are linear

2000/12/13 by Stephen Bigelow · 293 citations
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Algorithm #Annotation #Computer science #Artificial intelligence #Type (biology) #Mathematics #Biology

paper · pdf · doi:10.1090/s0894-0347-00-00361-1

published in Journal of the American Mathematical Society 14(2), 471-486 (American Mathematical Society)

openalex publication_date 2000/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

The braid group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript n"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be defined as the mapping class group of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -punctured disk. A group is said to be linear if it admits a faithful representation into a group of matrices over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 4"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper, we show that it is faithful for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

Citations

• Homological representations of the Hecke algebra
• On Krammer's Representation of the Braid Group

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• Polynomial-Time Solutions of Computational Problems in Noncommutative-Algebraic Cryptography
• Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type A_n-1, PhD thesis California Institute of Technology 2008
• The Pure Braid Groups and Their Relatives
• Automorphism groups of free groups, surface groups and free abelian groups
• Twisted homological stability for configuration spaces
• On extensions of the standard representation of the braid group to the singular braid group
• Period-Doubling Cascades Invariants: Braided Routes To Chaos
• Linear group [wikipedia]