nLab Yetter-Drinfeld module

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YetterDrinfeld modules

Yetter–Drinfeld modules

Definition

A Yetter–Drinfeld module over a kk-bialgebra B=(B,Δ,ϵ)B=(B,\Delta,\epsilon), (with Sweedler notation Δ(b)=b (1)b (2)\Delta(b) = \sum b_{(1)}\otimes b_{(2)}), is a kk-module which is simultaneously a BB-module and a BB-comodule with certain compatibility – also called Yetter-Drinfeld condition – between the BB-action and BB-coaction.

Compatibility for left-right YD Modules

The compatibility for a left BB-module BMMB\otimes M\to M, bmbmb\otimes m\mapsto b\blacktriangleright m, which is a right BB-comodule with respect to the coaction ρ:MMB\rho:M\to M\otimes B, ρ(m)=m [0]m [1]\rho(m) = \sum m_{[0]}\otimes m_{[1]}, is the following

(b (1)m [0])b (2)m [1]=(b (2)m) [0](b (2)m) [1]b (1) \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 BB is a Hopf algebra with invertible antipode SS (or instead just with the skew-antipode denoted S 1S^{-1})

(b (2)m [0])b (3)m [1]S 1(b (1))=(bm) [0](bm) [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](b (2)m [0])=(b (1)m) [1]b (2)(b (1)m) [0] 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)(m [0]b (2))=b (2)(mb (1)) [1](mb (1)) [0] 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]b (1)m [1]b (2)=(mb (2)) [0]b (1)(mb (2)) [1] 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 BB-modules are morphisms of underlying BB-modules which are also the morphisms of underlying BB-comodules. The category of left-right YD modules over a bialgebra BB is denoted by B𝒴𝒟 B{}_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.

B𝒴𝒟 B{}_B \mathcal{Y D}^B is a monoidal category: if VV and WW are left-right YD modules, VWV\otimes W is the tensor product of underlying vector spaces equipped with left BB-action

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

and right BB-coaction

vwv [0]w [0]w [1]v [1] 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 BB. (They mention, however, both structures.)

Monoidal category B𝒴𝒟 B{}_B \mathcal{Y D}^B is equipped with “pre-braiding” morphisms

R V,W:VWWV,vww [0](w [1]v). 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 vw(v [1]w)v [0]v\otimes w\mapsto (v_{[1]}\blacktriangleright w)\otimes v_{[0]}. Prebraidings satisfy all conditions for a braiding except for invertibility of R V,WR_{V,W} which is fullfilled for all V,WV,W iff BB is a Hopf algebra. R V,WR_{V,W} is always fullfilled if both VV and WW are finite dimensional. In particular, R V,VR_{V,V} satisfies the Yang-Baxter equation. If AA is a commutative algebra in B𝒴𝒟 B{}_B\mathcal{Y D}^B then the smash product algebra ABA\sharp B is an associative bialgebroid, said to be the extension of scalars from the bialgebra BB along kAk\hookrightarrow A. If B=HB=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=HB=H is a finite-dimensional Hopf algebra, then the category H𝒴𝒟 H{}_H \mathcal{Y D}^H is equivalent to the category of D(H){}_{D(H)}\mathcal{M} of left D(H)D(H)-modules, where D(H)D(H) is the Drinfeld double of HH, which in turn is equivalent to the center of the monoidal category H{}_H\mathcal{H} of left HH-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 H𝒴𝒟 H{}_H\mathcal{Y D}^H provide such examples. An important example, is the dual H *H^* when HH is finite-dimensional. The smash product algebra is in that case the Heisenberg double, hence it inherits a Hopf algebroid structure.

If FF is a counital 2-cocycle for a bialgebra HH, the Drinfeld twist H FH^F of FF is also a bialgebra and there is a monoidal equivalence H H F{}_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 H H F H F{}_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,μ)(A,\mu) in B𝒴𝒟 B{}_B\mathcal{Y D}^B. Let its multiplication map be denoted μ:acac\mu:a\otimes c\mapsto a\cdot c. Let us unwind the requirements that μ:AAA\mu:A\otimes A\to A is a morphism in B𝒴𝒟 B{}_B\mathcal{Y D}^B.

Requirement that μ\mu is a map of BB-modules is, for a,cAa,c\in A

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

which, together with compatibility of unit b1=ϵ(b)1b\blacktriangleright 1 = \epsilon(b) 1, means that the action is Hopf (AA is a left BB-module algebra). Requirement that μ\mu is a map of BB-comodules is

ρμ=(μid)ρ AA\rho\circ\mu = (\mu\otimes id)\rho_{A\otimes A}
ρ(ac)=(μid)(a [0]c [0]c [1]a [1])=a [0]c [0]c [1]a [1], \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), AA is right B opB^\op-comodule algebra. A left-right Yetter-Drinfeld module algebra AA is braided-commutative if

μR A,A=μ. \mu\circ R_{A,A} = \mu.

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

c [0](c [1]a)=ac. 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 HH-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 (× R\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

Last revised on October 20, 2024 at 11:28:41. See the history of this page for a list of all contributions to it.