Yetter–Drinfeld category

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 $\Delta$ 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

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

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

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

Examples

• Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction $\delta (v)=1\otimes v$.
• The trivial module $V=k\{v\}$ with $h{\boldsymbol {.}}v=\epsilon (h)v$, $\delta (v)=1\otimes v$, 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

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

where each $V_{g}$ 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

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

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

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

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

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

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

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

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

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

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

  (this notation emphasizes the graduation$h\otimes v\in V_{h}$, 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 $c_{{V,W}}:V\otimes W\to W\otimes V$,

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

is invertible with inverse

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

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

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

A monoidal category ${\mathcal {C}}$ 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 ${}_{H}^{H}{\mathcal {YD}}$.

References

1. N. Andruskiewitsch and M.Grana: Braided Hopf algebras over non abelian groups, Bol. Acad. Ciencias (Cordoba) 63(1999), 658-691

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