INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER
INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER
复制标题
德林菲尔德中心的内部自然变换和弗罗贝尼乌斯代数
DOI：
10.1007/s00031-021-09678-5
复制
发表时间：
2020
影响因子：
0.7
通讯作者：
C. Schweigert
中科院分区：
文献类型：
--
作者：
J. Fuchs;C. Schweigert
For ℳ and N $$ \mathcal{N} $$ finite module categories over a finite tensor category C $$ \mathcal{C} $$ , the category ℛ ex C $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ (ℳ, N $$ \mathcal{N} $$ ) of right exact module functors is a finite module category over the Drinfeld center Z $$ \mathcal{Z} $$ ( C $$ \mathcal{C} $$ ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C $$ \mathcal{C} $$ - C $$ \mathcal{C} $$ -bimodule functors to objects of Z $$ \mathcal{Z} $$ ( C $$ \mathcal{C} $$ ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and N $$ \mathcal{N} $$ are exact C $$ \mathcal{C} $$ -modules and C $$ \mathcal{C} $$ is pivotal, then the Z $$ \mathcal{Z} $$ ( C $$ \mathcal{C} $$ )-module ℛ ex C $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ (ℳ, N $$ \mathcal{N} $$ ) is exact. We compute its relative Serre functor and show that if ℳ and N $$ \mathcal{N} $$ are even pivotal module categories, then ℛ ex C $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ (ℳ, N $$ \mathcal{N} $$ ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in Z $$ \mathcal{Z} $$ ( C $$ \mathcal{C} $$ ).
For ℳ and N $$ \mathcal{N} $$ finite module categories over a finite tensor category C $$ \mathcal{C} $$ , the category ℛ ex C $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ (ℳ, N $$ \mathcal{N} $$ ) of right exact module functors is a finite module category over the Drinfeld center Z $$ \mathcal{Z} $$ ( C $$ \mathcal{C} $$ ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C $$ \mathcal{C} $$ - C $$ \mathcal{C} $$ -bimodule functors to objects of Z $$ \mathcal{Z} $$ ( C $$ \mathcal{C} $$ ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and N $$ \mathcal{N} $$ are exact C $$ \mathcal{C} $$ -modules and C $$ \mathcal{C} $$ is pivotal, then the Z $$ \mathcal{Z} $$ ( C $$ \mathcal{C} $$ )-module ℛ ex C $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ (ℳ, N $$ \mathcal{N} $$ ) is exact. We compute its relative Serre functor and show that if ℳ and N $$ \mathcal{N} $$ are even pivotal module categories, then ℛ ex C $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ (ℳ, N $$ \mathcal{N} $$ ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in Z $$ \mathcal{Z} $$ ( C $$ \mathcal{C} $$ ).