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/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Abstract  We tackle the problem of constructing    R  $R$   ‐matrices for the category    O  $\mathcal {O}$    associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebraU  q    (  g  )    $U_q({\mathfrak {g}})$   . For this, we define an invertible exact functorF  q   $\mathcal {F}_q$    from the category    O  $\mathcal {O}$    linked toU   q   −  1(  g  )    $U_{q^{-1}}({\mathfrak {g}})$    to the one linked toU  q    (  g  )    $U_q({\mathfrak {g}})$   . This functorF  q   $\mathcal {F}_q$    is compatible with tensor products, preserves irreducibility, and interchanges the subcategoriesO  +   $\mathcal {O}^+$    andO  −   $\mathcal {O}^-$    of Hernandez and Leclerc ( Algebra Number Theory   10  (2016) 2015–2052). We construct    R  $R$   ‐matrices forO  +   $\mathcal {O}^+$    by applyingF  q   $\mathcal {F}_q$    on the braidings already found forO  −   $\mathcal {O}^-$    by Hernandez ( Represent. Theory  26 (2022) 179–210). We also use the factorization of the latter intertwiners in terms of stable maps to deduce an analogous factorization for our new braidings. We finally obtain as byproducts new relations for the Grothendieck ringK  0    (  O  )    $K_0(\mathcal {O})$    as well as a functorial interpretation of a remarkable ring isomorphismK  0    (   O  +   )   ≃   K  0    (   O  −   )    $K_0(\mathcal {O}^+)\simeq K_0(\mathcal {O}^-)$    of Hernandez–Leclerc.

A functor for constructing R $R$ ‐matrices in the category O $\mathcal {O}$ of Borel quantum loop algebras