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The Braided Exterior Algebra

  • Stephen Bruce Sontz
Part of the Universitext book series (UTX)

Abstract

Using the representation ϕ k of the braid group B k , we next define (in some sort of analogy with the case of the symmetric group) an antisymmetrization operator
$$\displaystyle{A_{k}: \Gamma ^{\otimes k} \rightarrow \Gamma ^{\otimes k}}$$
for all k ≥ 2. (When you come to think of it, it would really be better called an antibraidization operator.) Then the elements of degree k ≥ 2 of the braided exterior algebra associated to \(\Gamma \) will be defined by
$$\displaystyle{ \wedge ^{k}\Gamma:= \Gamma ^{\otimes k}/\ker A_{ k}\mathop{\cong}\mathrm{Ran}\,A_{k}. }$$
(9.1)
Recall that we have already decided on the definitions \(\wedge ^{0}\Gamma:= \mathcal{A}\) as well as \(\wedge ^{1}\Gamma:= \Gamma \). These two definitions are equivalent to defining A0 and A1 to be the appropriate identity maps. So it comes down to defining A k —and understanding it—for k ≥ 2. Unless mentioned otherwise, we take k ≥ 2 throughout this section. Actually, some of the statements we will make are false for k = 1.

Bibliography

  1. 46.
    C. Kassel and V. Turaev, Braid Groups, Springer, 2008.Google Scholar
  2. 86.
    S.L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Commun. Math. Phys. 122 (1989) 125–170.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Stephen Bruce Sontz
    • 1
  1. 1.Centro de Investigación en Matemáticas, A.C.GuanajuatoMexico

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