nLab
Yetter-Drinfeld module

Definition

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 between the $B$-action and $B$-coaction.

Compatibility for left-right YD Modules

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

$$\sum (b_{(1)}\blacktriangleright m_{[0]})\otimes b_{(2)} m_{[1]} = \sum (b_{(2)}\blacktriangleright m)_{[0]} \otimes (b_{(2)}\blacktriangleright m)_{[1]} b_{(1)}$$

or equivalently, if $B$ is a Hopf algebra with invertible antipode $S$ (or instead just with the skew-antipode denoted $S^{-1}$)

$$\sum (b_{(2)}\blacktriangleright m_{[0]})\otimes b_{(3)} m_{[1]} S^{-1}(b_{(1)}) = \sum (b\blacktriangleright m)_{[0]} \otimes (b\blacktriangleright m)_{[1]}$$

Compatibility for left-left YD Modules

$$b_{(1)} m_{[-1]}\otimes (b_{(2)}\blacktriangleright m_{[0]}) = (b_{(1)}\blacktriangleright m)_{[-1]} b_{(2)} \otimes (b_{(1)}\blacktriangleright m)_{[0]}$$

Compatibility for right-left YD Modules

$$m_{[-1]}b_{(1)}\otimes (m_{[0]}\blacktriangleleft b_{(2)}) = b_{(2)} (m\blacktriangleleft b_{(1)})_{[-1]} \otimes (m\blacktriangleleft b_{(1)})_{[0]}$$

Compatibility for right-right YD Modules

$$m_{[0]}\blacktriangleleft b_{(1)}\otimes m_{[1]} b_{(2)} = (m\blacktriangleleft b_{(2)})_{[0]}\otimes b_{(1)} (m\blacktriangleleft b_{(2)})_{[1]}$$

The category of Yetter–Drinfeld modules

The 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.

Anti Yetter–Drinfeld 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. Brzeziński on YD modules, however their identification are not restricted to AYD and works for a wide variety of YD type modules.

Literature

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• A.M. Semikhatov, Yetter–Drinfeld structures on Heisenberg doubles and chains, arXiv:0908.3105

• wikipedia Yetter–Drinfeld category

• Peter Schauenburg, Hopf Modules and Yetter–Drinfel′d Modules, Journal of Algebra 169:3 (1994) 874–890 doi; Hopf modules and the double of a quasi-Hopf algebra, Trans. Amer. Math. Soc. 354 (2002), 3349–3378 doi pdf; Actions of monoidal categories, and generalized Hopf smash products, Journal of Algebra 270 (2003) 521–563, doi ps

• V. G. Drinfel’d, Quantum groups, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 798–820, AMS 1987, djvu:1.3M, pdf:2.5M

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• Gabriella Böhm, Dragos Stefan, (Co)cyclic (co)homology of bialgebroids: An approach via (co)monads, Comm. Math. Phys. 282 (2008), no.1, 239–286, arxiv/0705.3190; A categorical approach to cyclic duality, J. Noncommutative Geometry 6 (2012), no. 3, 481–538, arxiv/0910.4622

• Atabey Kaygun, Masoud Khalkhali, Hopf modules and noncommutative differential geometry, Lett. in Math. Physics 76:1, pp 77–91 (2006) arxiv/math.QA/0512031, doi

• T. Brzeziński, 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 arxiv:math.QA/0608170

• P.M. Hajac, M. Khalkhali, B. Rangipour, Y. Sommerhaeuser, Hopf-cyclic homology and cohomology with coefficients, C. R. Math. Acad. Sci. Paris 338(9), 667–672 (2004) math.KT/0306288; Stable anti-Yetter–Drinfeld modules. C. R. Math Acad. Sci. Paris 338(8), 587–590 (2004)

• B. Rangipour, Serkan Sütlü, Characteristic classes of foliations via SAYD-twisted cocycles, arxiv/1210.5969; SAYD modules over Lie–Hopf algebras, http://arxiv.org/abs/1108.6101; Cyclic cohomology of Lie algebras, arxiv/1108.2806

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• Georgia Benkart, Mariana Pereira, Sarah Witherspoon, Yetter–Drinfeld modules under cocycle twists, arxiv/0908.1563

• Shahn Majid, Robert Oeckl, Twisting of quantum differentials and

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