A Yetter–Drinfeld module over a -bialgebra , (with Sweedler notation ), is a -module which is simultaneously a -module and a -comodule with certain compatibility – also called Yetter-Drinfeld condition – between the -action and -coaction.
The compatibility for a left -module , , which is a right -comodule with respect to the coaction , , is the following
or equivalently, if is a Hopf algebra with invertible antipode (or instead just with the skew-antipode denoted )
Morphisms of YD -modules are morphisms of underlying -modules which are also the morphisms of underlying -comodules. The category of left-right YD modules over a bialgebra is denoted by ; 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.
is a monoidal category: if and are left-right YD modules, is the tensor product of underlying vector spaces equipped with left -action
and right -coaction
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 . (They mention, however, both structures.)
Monoidal category is equipped with “pre-braiding” morphisms
In Radford-Towber convention the pre-braiding is . Prebraidings satisfy all conditions for a braiding except for invertibility of which is fullfilled for all iff is a Hopf algebra. is always fullfilled if both and are finite dimensional. In particular, satisfies the Yang-Baxter equation. If is a commutative algebra in then the smash product algebra is an associative bialgebroid, said to be the extension of scalars from the bialgebra along . If 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 is a finite-dimensional Hopf algebra, then the category is equivalent to the category of of left -modules, where is the Drinfeld double of , which in turn is equivalent to the center of the monoidal category of left -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 provide such examples. An important example, is the dual when is finite-dimensional. The smash product algebra is in that case the Heisenberg double, hence it inherits a Hopf algebroid structure.
If is a counital 2-cocycle for a bialgebra , the Drinfeld twist of is also a bialgebra and there is a monoidal equivalence . 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 .
A left-right Yetter-Drinfeld module algebra is a monoid in . Let its multiplication map be denoted . Let us unwind the requirements that is a morphism in .
Requirement that is a map of -modules is, for
which, together with compatibility of unit , means that the action is Hopf ( is a left -module algebra). Requirement that is a map of -comodules is
that is (along with the counit condition), is right -comodule algebra. A left-right Yetter-Drinfeld module algebra is braided-commutative if
In explicit terms, for all ,
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.
To every YD module one assigns a Nichols algebra, see there.
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