The Braided Exterior Algebra

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 A ₀ and A ₁ 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.