English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In the present paper, we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representationsBn→GLn⁢(n-1)/2⁢(Z⁢[t±1])B_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1}]),VBn→GLn⁢(n-1)/2⁢(Z⁢[t±1,t1±1,t2±1,…,tn-1±1])\mathrm{VB}_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1},t_{1}^{\pm 1},t_{2}^{\pm 1},\ldots,t_{n-1}^{\pm 1}])which are connected with the famous Lawrence–Bigelow–Krammer representation. It turns out that these representations induce faithful representations of the crystallographic groupsBn/Pn′B_{n}/P_{n}^{\prime},VBn/VPn′\mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime}, respectively. Using these representations we study certain properties of the groupsBn/Pn′B_{n}/P_{n}^{\prime},VBn/VPn′\mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime}. Moreover, we construct new representations and decompositions of the universal braid groupsUBn\mathrm{UB}_{n}.

Virtual and universal braid groups, their quotients and representations