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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeMay 17th 2011
    • (edited May 19th 2011)

    The stub for “associative” bialgebroid. Bialgebroids are to bialgebras what on dual side groupoids are to groups. More references at Hopf algebroids.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeMay 19th 2011
    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeOct 2nd 2012
    • (edited Oct 2nd 2012)

    Some new material at bialgebroid, following

    All modules and morphisms will be over a fixed ground commutative ring k.

    A left A-bialgebroid is an AkAop-ring (H,μH,η), together with the A-bimodule map Δ:AHAH, which is coassociative and counital with counit ε, such that

    (i) the A-bimodule structure used on H is a.h.a:=s(a)t(a)h, where s:=η(1A):AH and t:=η(1A):AopH are the algebra maps induced by the unit η of the AAop-ring H

    (ii) the coproduct Δ:HHAH corestricts to the Takeuchi product and the corestriction Δ:HH×AH is a k-algebra map, where the Takeuchi product H×AH has a multiplication induced factorwise

    (iii) ε is a left character on the A-ring (H,μH,s)

    Notice that HAH is in general not an algebra, just an A-bimodule.

    The definition of a right A-bialgebroid differs by the A-bimodule structure on H given instead by a.h.a:=hs(a)t(a) and the counit ε is a right character on the A-coring (H,μH,t) (t and s can be interchanged in the last requirement).

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeDec 16th 2020

    I am writing an explicit definition of a left bialgebroid for the wikipedia in a different way than I previously put into the entry so I am copying a variant here to have yet another equivalent definition, avoiding AAop-rings.

    diff, v14, current

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeDec 16th 2020

    Minor fixes

    diff, v15, current

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeOct 31st 2021

    Formatting

    diff, v16, current