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On the representation theory of braid groups. (English. French summary) Zbl 1303.20047
This 67 pages long paper develops a technique to construct representations in a more systematic way and to study them. They are representations of certain groups related with the Artin braid groups \(B_n\) (the \(n\)-th braid group of the disk). As main application the author recovers several well known representations of \(B_n\) as well representations which factor through certain quotients of \(B_n\), among others the quotient of \(B_n\) by the commutator subgroup \([P_n,P_n]\) where \(P_n\) denotes the pure braid group on \(n\) strings (but not all). He also finds new representations. Many of the details are too specific and technical facts about representations to be stated here.
The main stream of the paper uses the fundamental article by V. G. Drinfel’d [Algebra Anal. 2, No. 4, 149-181 (1990; Zbl 0718.16034); translation in Leningr. Math. J. 2, No. 4, 829-860 (1991)] to obtain the set \(\text{Assoc}_\lambda(\mathbb K)\) of all formal series \(\Phi\) in two non-commuting variables \(A\) and \(B\), which satisfy certain relations. For any \(\Phi\in\text{Assoc}_\lambda(\mathbb K)\) by Drinfeld there exists a homomorphism from \(\mathbb KB_n\) to the completion \(\widehat{\mathcal B}_n\) of \(\mathcal B_n\). Here \(B_n\) denotes Artin’s braid group, \(\mathcal B_n\) the semi-direct product \(\mathbb Q\mathcal C_n\ltimes\cup\mathcal T_n\) and \(\mathcal T_n\) the Lie algebra of infinitesimal pure braids on \(n\) strands.
Once we have a representation \(\rho\colon\mathcal B_n\to M_N(\mathbb K)\) of \(\mathcal B_n\) over \(\mathbb K\) we may extend \(\rho\) to a representation \(\rho''\colon\widehat{\mathcal B}_n\to\mathbb K[[h]]\), where \(\mathbb K[[h]]\) is the completion of the polynomial ring \(\mathbb K[h]\) in one variable \(h\). After composing \(\rho''\) with the homomorphism associated with \(\Phi\) and taking the restriction to \(B_n\), we obtain a representation of \(B_n\) denoted by \(\widehat\Phi(\rho)\). The paper extensively studies this correspondence \(\rho\to\widehat\Phi(\rho)\). More specifically which properties of \(\rho\) are preserved under this correspondence like linearly independence, Tannakian, generically Tannakian, etc., whatever they mean. This general approach is used to recover several of the main classical representations. Notoriously the Iwahori-Hecke and Birman-Wenzl-Murakami algebras, Yang-Baxter representation, etc. The following sentence from the paper describes in a precise way the goal of the article: “We show how the representations of the braid groups which appeared in the last decades can be dealt with in our terms, and how our approach – in particular our uniform approach to unitary representations – may simplify or improve either the constructions or the proof of their main properties.”
Some considerations about convergence of the elements \(\Phi\) are made and they appear in the appendix A. The paper is well organized and requires a good background in representations from the reader.

20F36 Braid groups; Artin groups
20C15 Ordinary representations and characters
Full Text: DOI arXiv
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