In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.

Definition

Let H be a Hopf algebra over a field k. Let ${\displaystyle \Delta }$ denote the coproduct and S the antipode of H. Let V be a vector space over k. Then V is called a (left left) Yetter–Drinfeld module over H if

• ${\displaystyle (V,{\boldsymbol {.}})}$ is a left H-module, where ${\displaystyle {\boldsymbol {.}}:H\otimes V\to V}$ denotes the left action of H on V,
• ${\displaystyle (V,\delta \;)}$ is a left H-comodule, where ${\displaystyle \delta :V\to H\otimes V}$ denotes the left coaction of H on V,
• the maps ${\displaystyle {\boldsymbol {.}}}$ and ${\displaystyle \delta }$ satisfy the compatibility condition

${\displaystyle \delta (h{\boldsymbol {.}}v)=h_{(1)}v_{(-1)}S(h_{(3)})\otimes h_{(2)}{\boldsymbol {.}}v_{(0)}}$ for all ${\displaystyle h\in H,v\in V}$,

where, using Sweedler notation, ${\displaystyle (\Delta \otimes \mathrm {id} )\Delta (h)=h_{(1)}\otimes h_{(2)}\otimes h_{(3)}\in H\otimes H\otimes H}$ denotes the twofold coproduct of ${\displaystyle h\in H}$, and ${\displaystyle \delta (v)=v_{(-1)}\otimes v_{(0)}}$.

Examples

• Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction ${\displaystyle \delta (v)=1\otimes v}$.
• The trivial module ${\displaystyle V=k\{v\}}$ with ${\displaystyle h{\boldsymbol {.}}v=\epsilon (h)v}$, ${\displaystyle \delta (v)=1\otimes v}$, is a Yetter–Drinfeld module for all Hopf algebras H.
• If H is the group algebra kG of an abelian group G, then Yetter–Drinfeld modules over H are precisely the G-graded G-modules. This means that

${\displaystyle V=\bigoplus _{g\in G}V_{g}}$,

where each ${\displaystyle V_{g}}$ is a G-submodule of V.

• More generally, if the group G is not abelian, then Yetter–Drinfeld modules over H=kG are G-modules with a G-gradation

${\displaystyle V=\bigoplus _{g\in G}V_{g}}$, such that ${\displaystyle g.V_{h}\subset V_{ghg^{-1}}}$.

• Over the base field ${\displaystyle k=\mathbb {C} \;}$ all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given^[1] through a conjugacy class ${\displaystyle [g]\subset G\;}$ together with ${\displaystyle \chi ,X\;}$ (character of) an irreducible group representation of the centralizer ${\displaystyle Cent(g)\;}$ of some representing ${\displaystyle g\in [g]}$:

  ${\displaystyle V={\mathcal {O}}_{[g]}^{\chi }={\mathcal {O}}_{[g]}^{X}\qquad V=\bigoplus _{h\in [g]}V_{h}=\bigoplus _{h\in [g]}X}$

  • As G-module take ${\displaystyle {\mathcal {O}}_{[g]}^{\chi }}$ to be the induced module of ${\displaystyle \chi ,X\;}$:

  ${\displaystyle Ind_{Cent(g)}^{G}(\chi )=kG\otimes _{kCent(g)}X}$

  (this can be proven easily not to depend on the choice of g)

  • To define the G-graduation (comodule) assign any element ${\displaystyle t\otimes v\in kG\otimes _{kCent(g)}X=V}$ to the graduation layer:

  ${\displaystyle t\otimes v\in V_{tgt^{-1}}}$

  • It is very custom to directly construct ${\displaystyle V\;}$ as direct sum of X´s and write down the G-action by choice of a specific set of representatives ${\displaystyle t_{i}\;}$ for the ${\displaystyle Cent(g)\;}$-cosets. From this approach, one often writes

  ${\displaystyle h\otimes v\subset [g]\times X\;\;\leftrightarrow \;\;t_{i}\otimes v\in kG\otimes _{kCent(g)}X\qquad {\text{with uniquely}}\;\;h=t_{i}gt_{i}^{-1}}$

  (this notation emphasizes the graduation${\displaystyle h\otimes v\in V_{h}}$, rather than the module structure)

Braiding

Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map ${\displaystyle c_{V,W}:V\otimes W\to W\otimes V}$,

${\displaystyle c(v\otimes w):=v_{(-1)}{\boldsymbol {.}}w\otimes v_{(0)},}$

is invertible with inverse

${\displaystyle c_{V,W}^{-1}(w\otimes v):=v_{(0)}\otimes S^{-1}(v_{(-1)}){\boldsymbol {.}}w.}$

Further, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation

${\displaystyle (c_{V,W}\otimes \mathrm {id} _{U})(\mathrm {id} _{V}\otimes c_{U,W})(c_{U,V}\otimes \mathrm {id} _{W})=(\mathrm {id} _{W}\otimes c_{U,V})(c_{U,W}\otimes \mathrm {id} _{V})(\mathrm {id} _{U}\otimes c_{V,W}):U\otimes V\otimes W\to W\otimes V\otimes U.}$

A monoidal category ${\displaystyle {\mathcal {C}}}$ consisting of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is called a Yetter–Drinfeld category. It is a braided monoidal category with the braiding c above. The category of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is denoted by ${\displaystyle {}_{H}^{H}{\mathcal {YD}}}$.

References

• Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.