Formatting

]]>Minor fixes

]]>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 $A\otimes A^{op}$-rings.

]]>Some new material at bialgebroid, following

- Gabriella Böhm,
*Hopf algebroids*, (a chapter of) Handbook of algebra, arxiv:math.RA/0805.3806

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).

Related new stubs dynamical extension of a monoidal category and dynamical Yang-Baxter equation.

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

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