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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 $A\otimes_k A^{op}$-ring $(H,\mu_H,\eta)$, together with the $A$-bimodule map $\Delta : A\to H\otimes_A H$, which is coassociative and counital with counit $\epsilon$, such that
(i) the $A$-bimodule structure used on $H$ is $a.h.a':= s(a)t(a')h$, where $s := \eta(-\otimes 1_A):A\to H$ and $t:=\eta(1_A\otimes -):A^{op}\to H$ are the algebra maps induced by the unit $\eta$ of the $A\otimes A^{op}$-ring $H$
(ii) the coproduct $\Delta : H\to H\otimes_A H$ corestricts to the Takeuchi product and the corestriction $\Delta : H\to H\times_A H$ is a $k$-algebra map, where the Takeuchi product $H\times_A H$ has a multiplication induced factorwise
(iii) $\epsilon$ is a left character on the $A$-ring $(H,\mu_H,s)$
Notice that $H\otimes_A H$ 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 $a.h.a':= h s(a')t(a)$ and the counit $\epsilon$ is a right character on the $A$-coring $(H,\mu_H,t)$ ($t$ and $s$ can be interchanged in the last requirement).
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