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 – 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 $\rho:M\to M\otimes B$, $\rho(m) = \sum m_{[0]}\otimes m_{[1]}$, 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

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

$$v\otimes w\mapsto v_{[0]}\otimes w_{[0]}\otimes w_{[1]}v_{[1]}$$

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

$$R_{V,W}: V\otimes W\to W\otimes V,\,\,\,\,\,\,\,\, v\otimes w\mapsto w_{[0]} \otimes (w_{[1]}\blacktriangleright v).$$

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 ${}_H\mathcal{M}^H\cong {}_{H^F}\mathcal{M}^{H^F}$.

• Zoran Škoda, Martina Stojić, Comment on “Twisted bialgebroids versus bialgebroids from Drinfeld twist”, arXiv:2308.05083

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

$$\rho\circ\mu = (\mu\otimes id)\rho_{A\otimes A}$$

$$\rho(a\cdot c) = (\mu\otimes id)(a_{[0]}\otimes c_{[0]}\otimes c_{[1]} a_{[1]}) = a_{[0]}\cdot c_{[0]}\otimes c_{[1]} a_{[1]},$$

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 $a,c\in A$,

$$c_{[0]}\cdot (c_{[1]}\blacktriangleright a) = a\cdot c.$$

Anti Yetter–Drinfeld modules

The most general coefficients for Hopf cyclic cohomology are called stable anti-Yetter-Drinfeld modules; the definition of anti-Yetter-Drinfeld modules is very similar.

Nichols algebras

To every YD module one assigns a Nichols algebra, see there.

Literature

YD modules over Hopf/bi-algebras

• 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

• 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

• 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

• 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, Jacob 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

YD modules over Hopf/bi-algebroids

• Peter Schauenburg, Duals and doubles of quantum groupoids ($\times_R$-Hopf algebras), New Trends in Hopf Algebra Theory (La Falda, 1999), Contemp. Math. 267, Amer. Math. Soc. (2000) 273–299

• Imre Bálint, Kornél Szlachányi, Finitary Galois extensions over noncommutative bases, Journal of Algebra 296:2 (2006) 520–560 doi

• Xiao Han, Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids, arXiv:2312.03595

category: algebra