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\section*{Yetter-Drinfeld module}

\hypertarget{yetterdrinfeld_modules}{}\section*{{Yetter--Drinfeld modules}}\label{yetterdrinfeld_modules}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_category_of_yetterdrinfeld_modules}{The category of Yetter--Drinfeld modules}\dotfill \pageref*{the_category_of_yetterdrinfeld_modules} \linebreak
\noindent\hyperlink{anti_yetterdrinfeld_modules}{Anti Yetter--Drinfeld modules}\dotfill \pageref*{anti_yetterdrinfeld_modules} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{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 between the $B$-action and $B$-coaction.

\begin{udefn}
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

\begin{displaymath}
\sum (b_{(1)}\blacktriangleright m_{[0]})\otimes b_{(2)} m_{[1]}
= \sum (b_{(2)}\blacktriangleright m)_{[0]}
\otimes (b_{(2)}\blacktriangleright m)_{[1]} b_{(1)}
\end{displaymath}
or equivalently, if $B$ is a Hopf algebra with invertible antipode $S$ (or instead just with the [[skew-antipode]] denoted $S^{-1}$)

\begin{displaymath}
\sum (b_{(2)}\blacktriangleright m_{[0]})\otimes b_{(3)} m_{[1]} S^{-1}(b_{(1)})
= \sum (b\blacktriangleright m)_{[0]}
\otimes (b\blacktriangleright m)_{[1]}
\end{displaymath}
\end{udefn}
\begin{udefn}
\begin{displaymath}
b_{(1)} m_{[-1]}\otimes (b_{(2)}\blacktriangleright m_{[0]})
=
(b_{(1)}\blacktriangleright m)_{[-1]} b_{(2)}  \otimes 
(b_{(1)}\blacktriangleright m)_{[0]}
\end{displaymath}
\end{udefn}
\begin{udefn}
\begin{displaymath}
m_{[-1]}b_{(1)}\otimes (m_{[0]}\blacktriangleleft b_{(2)})
=
b_{(2)} (m\blacktriangleleft b_{(1)})_{[-1]}  \otimes (m\blacktriangleleft b_{(1)})_{[0]}
\end{displaymath}
\end{udefn}
\begin{udefn}
\begin{displaymath}
m_{[0]}\blacktriangleleft b_{(1)}\otimes m_{[1]} b_{(2)}
=
(m\blacktriangleleft b_{(2)})_{[0]}\otimes 
b_{(1)} (m\blacktriangleleft b_{(2)})_{[1]}
\end{displaymath}
\end{udefn}
\hypertarget{the_category_of_yetterdrinfeld_modules}{}\subsection*{{The category of Yetter--Drinfeld modules}}\label{the_category_of_yetterdrinfeld_modules}

The \emph{category of left-right YD modules}, i.e. Yetter--Drinfeld modules which are left $B$-modules and right $B$-comodules, 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 equipped with ``pre-braiding'' morphisms, which make it into a [[braided monoidal category]] iff $B$ is a Hopf algebra with a bijective antipode. 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.

\hypertarget{anti_yetterdrinfeld_modules}{}\subsection*{{Anti Yetter--Drinfeld modules}}\label{anti_yetterdrinfeld_modules}

The most general coefficients for Hopf cyclic cohomology is called stable-anti-Yetter--Drinfled 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. Brzeziski on YD modules, however their identification are not restricted to AYD and works for a wide variety of YD type modules.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\begin{itemize}%
\item Susan Montgomery, \emph{Hopf algebras and their actions on rings}, CBMS Lecture Notes \textbf{82}, AMS 1993, 240p.


\item A.M. Semikhatov, \emph{Yetter--Drinfeld structures on Heisenberg doubles and chains}, \href{http://arxiv.org/abs/0908.3105}{arXiv:0908.3105}


\item wikipedia \href{http://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld_category}{Yetter--Drinfeld category}


\item Peter Schauenburg, \emph{Hopf Modules and Yetter--Drinfel$\prime$d Modules}, Journal of Algebra \textbf{169}:3 (1994) 874--890 \href{http://dx.doi.org/10.1006/jabr.1994.1314}{doi}; \emph{Hopf modules and the double of a quasi-Hopf algebra}, Trans. Amer. Math. Soc. 354 (2002), 3349--3378 \href{http://dx.doi.org/10.1090/S0002-9947-02-02980-X}{doi} \href{http://www.ams.org/journals/tran/2002-354-08/S0002-9947-02-02980-X/S0002-9947-02-02980-X.pdf}{pdf}; \emph{Actions of monoidal categories, and generalized Hopf smash products}, Journal of Algebra \textbf{270} (2003) 521--563, \href{http://dx.doi.org/10.1016/S0021-8693%2803%2900403-4}{doi} \href{http://www.mathematik.uni-muenchen.de/%7Eschauen/papers/amcghsp.ps}{ps}


\item V. G. Drinfel'd, \emph{Quantum groups}, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 798--820, AMS 1987, \href{http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.djvu}{djvu:1.3M}, \href{http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf}{pdf:2.5M}


\item David N. Yetter, \emph{Quantum groups and representations of monoidal categories}, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 261--290 \href{http://www.ams.org/mathscinet-getitem?mr=1074714}{MR91k:16028} \href{http://dx.doi.org/10.1017/S0305004100069139}{doi}


\item Gabriella B\"o{}hm, Dragos Stefan, \emph{(Co)cyclic (co)homology of bialgebroids: An approach via (co)monads}, Comm. Math. Phys. 282 (2008), no.1, 239--286, \href{http://arxiv.org/abs/0705.3190}{arxiv/0705.3190}; \emph{A categorical approach to cyclic duality}, J. Noncommutative Geometry \textbf{6} (2012), no. 3, 481--538, \href{http://arxiv.org/abs/0910.4622}{arxiv/0910.4622}


\item Atabey Kaygun, Masoud Khalkhali, \emph{Hopf modules and noncommutative differential geometry}, Lett. in Math. Physics \textbf{76:1}, pp 77--91 (2006) \href{http://arxiv.org/abs/math/0512031}{arxiv/math.QA/0512031}, \href{http://dx.doi.org/10.1007/s11005-006-0062-x}{doi}


\item [[T. Brzeziński]], \emph{Flat connections and (co)modules}, in: New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35--52 \href{http://arxiv.org/abs/math.QA/0608170}{arxiv:math.QA/0608170}


\item P.M. Hajac, M. Khalkhali, B. Rangipour, Y. Sommerhaeuser, \emph{Hopf-cyclic homology and cohomology with coefficients}, C. R. Math. Acad. Sci. Paris \textbf{338}(9), 667--672 (2004) \href{http://arxiv.org/abs/math/0306288}{math.KT/0306288}; \emph{Stable anti-Yetter--Drinfeld modules}. C. R. Math Acad. Sci. Paris \textbf{338}(8), 587--590 (2004)


\item B. Rangipour, Serkan S\"u{}tl\"u{}, \emph{Characteristic classes of foliations via SAYD-twisted cocycles}, \href{http://arxiv.org/abs/1210.5969}{arxiv/1210.5969}; \emph{SAYD modules over Lie--Hopf algebras}, \href{http://arxiv.org/abs/1108.6101}{http://arxiv.org/abs/1108.6101}; \emph{Cyclic cohomology of Lie algebras}, \href{http://arxiv.org/abs/1108.2806}{arxiv/1108.2806}


\item Florin Panaite, Mihai D. Staic, \emph{Generalized (anti) Yetter--Drinfeld modules as components of a braided T-category}, \href{http://arxiv.org/abs/math/0503413}{math.QA/0503413}


\item D. Bulacu, S. Caenepeel, F. Panaite, \emph{Doi--Hopf modules and Yetter--Drinfeld categories for quasi-Hopf algebras}, Communications in Algebra, 34 (9), 3413--3449 (2006) \href{http://arxiv.org/abs/math/0311379}{math.QA/0311379}


\item Florin Panaite, Dragos Stefan, \emph{Deformation cohomology for Yetter--Drinfel'd modules and Hopf (bi)modules}, \href{http://arxiv.org/abs/math/0006048}{math.QA/0006048}


\item Nicol\'a{}s Andruskiewitsch, Istv\'a{}n Heckenberger, Hans-J\"u{}rgen Schneider, \emph{The Nichols algebra of a semisimple Yetter--Drinfeld module}, American J. of Math. \textbf{132}:6, (2010) 1493--1547 \href{http://dx.doi.org/10.1353/ajm.2010.0019}{doi}


\item M. Cohen, D. Fischman, S. Montgomery, \emph{On Yetter--Drinfeld categories and $H$-commutativity}, Commun. Algebra \textbf{27} (1999) 1321--1345


\item Yukio Doi, \emph{Hopf modules in Yetter--Drinfeld categories}, Commun. Alg. \textbf{26}:9, 3057--3070 (1998) \href{http://dx.doi.org/10.1080/00927879808826327}{doi}


\item I. Heckenberger, H.-J. Schneider, \emph{Yetter--Drinfeld modules over bosonizations of dually paired Hopf algebras}, \href{http://arxiv.org/abs/1111.4673}{arxiv/1111.4673}


\item V. Ulm, \emph{Actions of Hopf algebras in categories of Yetter--Drinfeld modules}, Comm. Alg. \textbf{31}:6, 2719--2743


\item D.E. Radford, J. Towber, \emph{Yetter--Drinfel'd categories associated to an arbitrary bialgebra}, J. Pure Appl. Algebra \textbf{87} (1993), 259--279 \href{http://www.ams.org/mathscinet-getitem?mr=1228157}{MR94f:16060} \href{http://dx.doi.org/10.1016/0022-4049%2893%2990114-9}{doi}


\item Georgia Benkart, Mariana Pereira, Sarah Witherspoon, \emph{Yetter--Drinfeld modules under cocycle twists}, \href{http://arxiv.org/abs/0908.1563}{arxiv/0908.1563}


\item [[Shahn Majid]], Robert Oeckl, \emph{Twisting of quantum differentials and the Planck scale Hopf algebra}, Commun. Math. Phys. \textbf{205}, 617--655 (1999)



\end{itemize}
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[[!redirects Yetter-Drinfeld category]] [[!redirects Yetter-Drinfeld categories]] [[!redirects Yetter–Drinfeld category]] [[!redirects Yetter–Drinfeld categories]] [[!redirects Yetter--Drinfeld category]] [[!redirects Yetter--Drinfeld categories]]



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