The Braided Exterior Algebra

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


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
$$\displaystyle{ \wedge ^{k}\Gamma:= \Gamma ^{\otimes k}/\ker A_{ k}\mathop{\cong}\mathrm{Ran}\,A_{k}. }$$
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.


  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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