nLab Yetter-Drinfeld module

YetterDrinfeld modules

Yetter–Drinfeld modules


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. 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.


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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.