Comparing self-dual corings, Frobenius corings and Ringel self-duality

2026/08/04 by Tomasz Brzeziński, Teresa Conde, Steffen Koenig +1
Mathematics · #math.RT #msc:16D90 #msc:16L60 #msc:16T15 #msc:17B10 #msc:18G25

paper · pdf

29 pages

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

Self-dual corings and self-dual algebra extensions are classified and these (self-)dualities are compared with Ringel (self-)duality, revealing fundamental differences. To each coring \mathcal C, two algebras are associated, known as the left and the right dual algebra of \mathcal C. It is shown that these two algebras coincide in a natural way if and only if \mathcal C is a Frobenius coring. Various other equivalent characterisations of \mathcal C being self-dual are given, in terms of certain algebra extensions being Frobenius extensions and in terms of certain forgetful or restriction functors being Frobenius functors. Ringel self-duality however is shown to be rather different, for which a homological explanation is given.

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