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2008 | Buch

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Braid Groups

verfasst von: Christian Kassel, Vladimir Turaev

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Braids and braid groups, the focus of this text, have been at the heart of important mathematical developments over the last two decades. Their association with permutations has led to their presence in a number of mathematical fields and physics. As central objects in knot theory and 3-dimensional topology, braid groups has led to the creation of a new field called quantum topology.

In this well-written presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence; this is followed by a treatment of the relationship between braids, knots and links. Important results then treat the linearity and orderability of the subject. Relevant additional material is included in five large appendices.

Braid Groups will serve graduate students and a number of mathematicians coming from diverse disciplines.

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Inhaltsverzeichnis

Frontmatter

1. Braids and Braid Groups

In this chapter we discuss the basics of the theory of braids and braid groups.

Christian Kassel, Vladimir Turaev

2. Braids, Knots, and Links

In this chapter we study the relationship between braids, knots, and links. Throughout the chapter, we denote by I the closed interval [0, 1] in R.

Christian Kassel, Vladimir Turaev

3. Homological Representations of the Braid Groups

Braid groups, viewed as the groups of isotopy classes of self-homeomorphisms of punctured disks, naturally act on the homology of topological spaces obtained from the punctured disks by functorial constructions. We discuss here two such constructions and study the resulting linear representations of the braid groups: the Burau representation (Sections 3.1–3.3) and the Lawrence–Krammer–Bigelow representation (Sections 3.5–3.7). As an application of the Burau representation, we construct in Section 3.3 the one-variable Alexander–Conway polynomial of links in R ³. As an application of the Lawrence–Krammer–Bigelow representation, we establish the linearity of B _n for all n (Section 3.5.4).

Christian Kassel, Vladimir Turaev

4. Symmetric Groups and Iwahori–Hecke Algebras

The study of the braid group B _n naturally leads to the so-called Iwahori—Hecke algebra H _n. This algebra is a finite-dimensional quotient of the group algebra of B _n depending on two parameters q and z. Our interest in the Iwahori—Hecke algebras is due to their connections to braids and links and to their beautiful representation theory discussed in the next chapter.

Christian Kassel, Vladimir Turaev

5. Representations of the Iwahori–Hecke Algebras

In this chapter we study the linear representations of the one-parameter Iwahori–Hecke algebras of Section 4.2.2. Our aim is to classify their finite-dimensional representations over an algebraically closed field of characteristic zero in terms of partitions and Young diagrams. As an application, we prove that the reduced Burau representation introduced in Section 3.3 is irreducible. We end the chapter by a discussion of the Temperley–Lieb algebras.

Christian Kassel, Vladimir Turaev

6. Garside Monoids and Braid Monoids

Braid groups may be viewed as groups of fractions of certain monoids called braid monoids. The latter belong to a wider class of so-called Garside monoids. In this chapter we investigate properties of monoids and specifically of Garside monoids. As an application, we give a solution of the conjugacy problem in the braid groups. We also discuss generalized braid groups associated with Coxeter matrices.

Christian Kassel, Vladimir Turaev

7. An Order on the Braid Groups

The principal aim of this chapter is to show that the braid groups have a natural total order.

Christian Kassel, Vladimir Turaev

A. Presentations of SL2(Z) and PSL2(Z)

Let \(\textrm{SL}_2({\bf Z})\) be the group of \(2 \times 2\) matrices with entries in \({\bf Z}\) and with determinant 1. The center of \(\textrm{SL}_2({\bf Z})\) is the group of order 2 generated by the scalar matrix \(-I_2\), where I ₂ is the unit matrix. The quotient group

$$\textrm{PSL}_2({\bf Z}) = \textrm{SL}_2({\bf Z})/\langle -I_2 \rangle$$

is called the modular group; it can be identified with the group of rational functions on \({\bf C}\) of the form \((az+b)/(cz+d)\), where a, b, c, d are integers such that \(ad - bc = 1\).

Christian Kassel, Vladimir Turaev

B. Fibrations and Homotopy Sequences

We recall several basic notions from the theory of fibrations needed in the main text. For details, the reader is referred, for instance, to [FR84, Chap. 5].

Christian Kassel, Vladimir Turaev

C. The Birman–Murakami–Wenzl Algebras

We briefly discuss a family of finite-dimensional quotients of the braid group algebras due to J. Murakami, J. Birman, and H. Wenzl. We also outline an interpretation of the Lawrence—Krammer—Bigelow representation of Section 3.5 in terms of representations of these algebras.

Christian Kassel, Vladimir Turaev

D. Left Self-Distributive Sets

We give here a brief introduction to so-called left self-distributive sets, which are closely related to braid groups.

Christian Kassel, Vladimir Turaev

Backmatter

Titel: Braid Groups
verfasst von: Christian Kassel
Vladimir Turaev
Copyright-Jahr: 2008
Verlag: Springer New York
Electronic ISBN: 978-0-387-68548-9
Print ISBN: 978-0-387-33841-5
DOI: https://doi.org/10.1007/978-0-387-68548-9

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