# nLab Yetter-Drinfeld module

YetterDrinfeld modules

# Yetter–Drinfeld modules

## 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 – also called Yetter-Drinfeld condition – 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 $\rho:M\to M\otimes B$, $\rho(m) = \sum m_{}\otimes m_{}$, is the following

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

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

$\sum (b_{(2)}\blacktriangleright m_{})\otimes b_{(3)} m_{} S^{-1}(b_{(1)}) = \sum (b\blacktriangleright m)_{} \otimes (b\blacktriangleright m)_{}$
###### Compatibility for left-left YD Modules
$b_{(1)} m_{[-1]}\otimes (b_{(2)}\blacktriangleright m_{}) = (b_{(1)}\blacktriangleright m)_{[-1]} b_{(2)} \otimes (b_{(1)}\blacktriangleright m)_{}$
###### Compatibility for right-left YD Modules
$m_{[-1]}b_{(1)}\otimes (m_{}\blacktriangleleft b_{(2)}) = b_{(2)} (m\blacktriangleleft b_{(1)})_{[-1]} \otimes (m\blacktriangleleft b_{(1)})_{}$
###### Compatibility for right-right YD Modules
$m_{}\blacktriangleleft b_{(1)}\otimes m_{} b_{(2)} = (m\blacktriangleleft b_{(2)})_{}\otimes b_{(1)} (m\blacktriangleleft b_{(2)})_{}$

## The category of Yetter–Drinfeld modules

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

$b\blacktriangleright (v\otimes w) = (b_{(1)}\blacktriangleright v)\otimes (b_{(2)}\blacktriangleright w)$

and right $B$-coaction

$v\otimes w\mapsto v_{}\otimes w_{}\otimes w_{}v_{}$

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

$R_{V,W}: V\otimes W\to W\otimes V,\,\,\,\,\,\,\,\, v\otimes w\mapsto w_{} \otimes (w_{}\blacktriangleright v).$

In Radford-Towber convention the pre-braiding is $v\otimes w\mapsto (v_{}\blacktriangleright w)\otimes v_{}$. 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 ${}_H\mathcal{M}^H\cong {}_{H^F}\mathcal{M}^{H^F}$.

## Yetter-Drinfeld module algebras

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 $a,c\in A$

$b\blacktriangleright (a\cdot c) = (b_{(1)}\blacktriangleright a)\cdot (b_{(2)}\blacktriangleright c),$

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

$\rho\circ\mu = (\mu\otimes id)\rho_{A\otimes A}$
$\rho(a\cdot c) = (\mu\otimes id)(a_{}\otimes c_{}\otimes c_{} a_{}) = a_{}\cdot c_{}\otimes c_{} a_{},$

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

$\mu\circ R_{A,A} = \mu.$

In explicit terms, for all $a,c\in A$,

$c_{}\cdot (c_{}\blacktriangleright a) = a\cdot c.$

## Anti Yetter–Drinfeld modules

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.

## Literature

• Susan Montgomery, Hopf algebras and their actions on rings, CBMS Lecture Notes 82, AMS 1993, 240p.

• 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

• David N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 261–290 MR91k:16028 doi

• 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, arXiv:1108.6101; Cyclic cohomology of Lie algebras, arXiv:1108.2806

• Florin Panaite, Mihai D. Staic, Generalized (anti) Yetter–Drinfeld modules as components of a braided T-category, arXiv:math.QA/0503413

• D. Bulacu, S. Caenepeel, F. Panaite, Doi–Hopf modules and Yetter–Drinfeld categories for quasi-Hopf algebras, Communications in Algebra, 34 (9), 3413–3449 (2006) math.QA/0311379

• Florin Panaite, Dragos Stefan, Deformation cohomology for Yetter–Drinfel’d modules and Hopf (bi)modules, math.QA/0006048

• Nicolás Andruskiewitsch, István Heckenberger, Hans-Jürgen Schneider, The Nichols algebra of a semisimple Yetter–Drinfeld module, American J. of Math. 132:6, (2010) 1493–1547 doi

• M. Cohen, D. Fischman, S. Montgomery, On Yetter–Drinfeld categories and $H$-commutativity, Commun. Algebra 27 (1999) 1321–1345

• Yukio Doi, Hopf modules in Yetter–Drinfeld categories, Commun. Alg. 26:9, 3057–3070 (1998) doi

• I. Heckenberger, H.-J. Schneider, Yetter–Drinfeld modules over bosonizations of dually paired Hopf algebras, arXiv:1111.4673

• V. Ulm, Actions of Hopf algebras in categories of Yetter–Drinfeld modules, Comm. Alg. 31:6, 2719–2743

• L. A. Lambe, D. E. Radford, Algebraic aspects of the quantum Yang–Baxter equation, J. Algebra 154 (1992) 228–288 doi

• D.E. Radford, J. Towber, Yetter–Drinfel’d categories associated to an arbitrary bialgebra, J. Pure Appl. Algebra 87 (1993), 259–279 MR94f:16060 doi

• Georgia Benkart, Mariana Pereira, Sarah Witherspoon, Yetter–Drinfeld modules under cocycle twists, J. Algebra 324:11 (2010) 2990–3006 arxiv:0908.1563

• Shahn Majid, Robert Oeckl, Twisting of quantum differentials and the Planck scale Hopf algebra, Commun. Math. Phys. 205, 617–655 (1999)

• Huixiang Chen, Yinhuo Zhang, Cocycle deformations and Brauer groups, Comm. Alg. 35:2 (2007) 399–433 doi; arXiv v. Cocycle deformation and Brauer group isomorphisms, arXiv:math/0505003

category: algebra

Last revised on August 10, 2023 at 09:05:01. See the history of this page for a list of all contributions to it.