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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
The stub for “associative” bialgebroid. Bialgebroids are to bialgebras what on dual side groupoids are to groups. More references at Hopf algebroids.
Related new stubs dynamical extension of a monoidal category and dynamical Yang-Baxter equation.
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 A⊗kAop-ring (H,μH,η), together with the A-bimodule map Δ:A→H⊗AH, 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):A→H and t:=η(1A⊗−):Aop→H are the algebra maps induced by the unit η of the A⊗Aop-ring H
(ii) the coproduct Δ:H→H⊗AH corestricts to the Takeuchi product and the corestriction Δ:H→H×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 H⊗AH 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).
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