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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We introduce and study new categories    T   g  ,  kof integrable    g  =  sl  (  ∞  )   ‐modules which depend on the choice of a certain reductive in   g   subalgebra    k  ⊂  g   . The simple objects of    T   g  ,  kare tensor modules as in the previously studied category    T  g    [Dan‐Cohen, Penkov and Serganova,  Adv. Math . 289 (2016) 250–278]; however, the choice of   k   provides for more flexibility of nonsimple modules in    T   g  ,  kcompared to    T  g   . We then choose   k   to have two infinite‐dimensional diagonal blocks, and show that a certain injective object    K   m  |  nin    T   g  ,  krealizes a categorical    sl  (  ∞  )   ‐action on the category    O   m  |  n   Z   , the integral category   O   of the Lie superalgebra    gl  (  m  |  n  )   . We show that the socle of    K   m  |  nis generated by the projective modules in    O   m  |  n   Z   , and compute the socle filtration of    K   m  |  nexplicitly. We conjecture that the socle filtration of    K   m  |  nreflects a ‘degree of atypicality filtration’ on the category    O   m  |  n   Z   . We also conjecture that a natural tensor filtration on    K   m  |  narises via the Duflo–Serganova functor sending the category    O   m  |  n   Z    to    O   m  −  1  |  n  −  1   Z   . We prove a weaker version of this latter conjecture for the direct summand of    K   m  |  ncorresponding to finite‐dimensional    gl  (  m  |  n  )   ‐modules.

Integrable sl ( ∞ ) ‐modules and category O for gl ( m | n )