A Yetter–Drinfeld module over a $k$-bialgebra $B=(B,\Delta,\epsilon)$, (with Sweedler notation $\Delta(b) = \sum b_{(1)}\otimes b_{(2)}$), is a $k$-module which is simultaneously a $B$-module and a $B$-comodule with certain compatibility – also called Yetter-Drinfeld condition – between the $B$-action and $B$-coaction.
The compatibility for a left $B$-module $B\otimes M\to M$, $b\otimes m\mapsto b\blacktriangleright m$, which is a right $B$-comodule with respect to the coaction $\rho:M\to M\otimes B$, $\rho(m) = \sum m_{[0]}\otimes m_{[1]}$, is the following
or equivalently, if $B$ is a Hopf algebra with invertible antipode $S$ (or instead just with the skew-antipode denoted $S^{-1}$)
Morphisms of YD $B$-modules are morphisms of underlying $B$-modules which are also the morphisms of underlying $B$-comodules. The category of left-right YD modules over a bialgebra $B$ is denoted by ${}_B \mathcal{Y D}^B$; the category is rarely alternatively called the (left-right) Yetter–Drinfeld category and it can be presented as the category of entwined modules for certain special entwining structure.
${}_B \mathcal{Y D}^B$ is a monoidal category: if $V$ and $W$ are left-right YD modules, $V\otimes W$ is the tensor product of underlying vector spaces equipped with left $B$-action
and right $B$-coaction
Note the order within the rightmost tensor factor! One checks directly that this tensor product indeed satisfies the Yetter-Drinfeld condition. Radford and Towber prefer slightly different monoidal structure: in above formulas use the opposite product and coopposite coproduct on $B$. (They mention, however, both structures.)
Monoidal category ${}_B \mathcal{Y D}^B$ is equipped with “pre-braiding” morphisms
In Radford-Towber convention the pre-braiding is $v\otimes w\mapsto (v_{[1]}\blacktriangleright w)\otimes v_{[0]}$. Prebraidings satisfy all conditions for a braiding except for invertibility of $R_{V,W}$ which is fullfilled for all $V,W$ iff $B$ is a Hopf algebra. $R_{V,W}$ is always fullfilled if both $V$ and $W$ are finite dimensional. In particular, $R_{V,V}$ satisfies the Yang-Baxter equation. If $A$ is a commutative algebra in ${}_B\mathcal{Y D}^B$ then the smash product algebra $A\sharp B$ is an associative bialgebroid, said to be the extension of scalars from the bialgebra $B$ along $k\hookrightarrow A$. If $B=H$ is a Hopf algebra with bijective antipode then this bialgebroid is in fact a Hopf algebroid, both in the sense of Lu and in the sense of Bohm.
If $B=H$ is a finite-dimensional Hopf algebra, then the category ${}_H \mathcal{Y D}^H$ is equivalent to the category of ${}_{D(H)}\mathcal{M}$ of left $D(H)$-modules, where $D(H)$ is the Drinfeld double of $H$, which in turn is equivalent to the center of the monoidal category ${}_H\mathcal{H}$ of left $H$-modules.
The commutative algebras in the center of a monoidal category, play role in the dynamical extension of a monoidal category. Hence the commutative algebras in ${}_H\mathcal{Y D}^H$ provide such examples. An important example, is the dual $H^*$ when $H$ is finite-dimensional. The smash product algebra is in that case the Heisenberg double, hence it inherits a Hopf algebroid structure.
If $F$ is a counital 2-cocycle for a bialgebra $H$, the Drinfeld twist $H^F$ of $F$ is also a bialgebra and there is a monoidal equivalence ${}_H\mathcal{M}\cong {}_{H^F}\mathcal{M}$. In Section 2 of Škoda-Stojić2023 it is shown how this monoidal equivalence lifts to a braided monoidal equivalence between the categories of Yetter-Drinfeld modules ${}_H\mathcal{M}^H\cong {}_{H^F}\mathcal{M}^{H^F}$.
A left-right Yetter-Drinfeld module algebra is a monoid $(A,\mu)$ in ${}_B\mathcal{Y D}^B$. Let its multiplication map be denoted $\mu:a\otimes c\mapsto a\cdot c$. Let us unwind the requirements that $\mu:A\otimes A\to A$ is a morphism in ${}_B\mathcal{Y D}^B$.
Requirement that $\mu$ is a map of $B$-modules is, for $a,c\in A$
which, together with compatibility of unit $b\blacktriangleright 1 = \epsilon(b) 1$, means that the action is Hopf ($A$ is a left $B$-module algebra). Requirement that $\mu$ is a map of $B$-comodules is
that is (along with the counit condition), $A$ is right $B^\op$-comodule algebra. A left-right Yetter-Drinfeld module algebra $A$ is braided-commutative if
In explicit terms, for all $a,c\in A$,
The most general coefficients for Hopf cyclic cohomology are called stable anti-Yetter-Drinfeld modules. These kind of modules appeared for the first time in different name in B. Rangipour’s PhD thesis under supervision of M. Khalkhali. Later on it was generalized by P.M. Hajac, M. Khalkhali, B. Rangipour, and Y. Sommerhaeuser. The category of AYD modules is not monodical but product of an AYD module with a YD module results in an AYD module. By the work of Rangipour–Sutlu one knows that there is such category over Lie algebras and there is a one-to-one correspondence between AYD modules over a Lie algebra and those over the universal enveloping algebra of the Lie algebra. This correspondence is extended by the same authors for bicrossed product Hopf algebras. The true meaning of the AYD modules in non commutative geometry is not known yet. There are some attempts by A. Kaygun–M. Khalkhali to relate them to the curvature of flat connections similar to the work of T. Brzeziński on YD modules, however their identification are not restricted to AYD and works for a wide variety of YD type modules.
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