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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 kk.

    A left AA-bialgebroid is an A kA opA\otimes_k A^{op}-ring (H,μ H,η)(H,\mu_H,\eta), together with the AA-bimodule map Δ:AH AH\Delta : A\to H\otimes_A H, which is coassociative and counital with counit ε\epsilon, such that

    (i) the AA-bimodule structure used on HH is a.h.a:=s(a)t(a)ha.h.a':= s(a)t(a')h, where s:=η(1 A):AHs := \eta(-\otimes 1_A):A\to H and t:=η(1 A):A opHt:=\eta(1_A\otimes -):A^{op}\to H are the algebra maps induced by the unit η\eta of the AA opA\otimes A^{op}-ring HH

    (ii) the coproduct Δ:HH AH\Delta : H\to H\otimes_A H corestricts to the Takeuchi product and the corestriction Δ:HH× AH\Delta : H\to H\times_A H is a kk-algebra map, where the Takeuchi product H× AHH\times_A H has a multiplication induced factorwise

    (iii) ε\epsilon is a left character on the AA-ring (H,μ H,s)(H,\mu_H,s)

    Notice that H AHH\otimes_A H is in general not an algebra, just an AA-bimodule.

    The definition of a right AA-bialgebroid differs by the AA-bimodule structure on HH given instead by a.h.a:=hs(a)t(a)a.h.a':= h s(a')t(a) and the counit ε\epsilon is a right character on the AA-coring (H,μ H,t)(H,\mu_H,t) (tt and ss 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 AA opA\otimes A^{op}-rings.

    diff, v14, current

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeDec 16th 2020

    Minor fixes

    diff, v15, current

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeOct 31st 2021


    diff, v16, current