Ryan Aziz | Searcharxiv

arXiv2025

Generalized Yetter-Drinfeld modules, the center of bi-actegories and groupoid-crossed braided bicategories

We study the notion of the $E$-center $\mathcal{Z}_E(\mathcal{M})$ of a $(\mathcal{C}, \mathcal{D})$-biactegory (or bimodule category) $\mathcal{M}$, relative to an op-monoidal functor $E: \mathcal{C} \to \mathcal{D}$. Specializing this notion to the case $\mathcal{M} = {}_A\mathrm{Mod}$, $\mathcal{C}={}_H\mathrm{Mod}$, $\mathcal{D} = {}_K\mathrm{Mod}$, and $E \simeq C\otimes_H - : {}_H\mathrm{Mod} \to {}_K\mathrm{Mod}$, where $H$ and $K$ are bialgebras, $A$ is an $(H,K)$-bicomodule algebra and $C$ is a $(K,H)$-bimodule coalgebra, we show that this $E$-center is equivalent to the category of generalized Yetter-Drinfeld modules as introduced by Caenepeel, Militaru, and Zhu. We introduce the notion of a double groupoid-crossed braided bicategory, generalizing Turaev's group-crossed braided monoidal categories, and show that generalized Yetter-Drinfeld modules can be organized in a double groupoid-crossed braided bicategory over the groupoids of Galois objects and co-objects.

math.RA↗

arXiv2019

Quantum differentials on cross product Hopf algebras

We construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product Hopf algebras, namely double cross products $A\hookrightarrow A\bowtie H\hookleftarrow H$, double cross coproducts $A\twoheadleftarrow A {\blacktriangleright\!\!\blacktriangleleft} H\twoheadrightarrow H$, biproducts $A{\buildrel\hookrightarrow\over \twoheadleftarrow}A{\cdot\kern-.33em\triangleright\!\!\!<} B$ and bicrossproducts $A\hookrightarrow A{\blacktriangleright\!\!\triangleleft} H\twoheadrightarrow H$ on the assumption that the factors have strongly bicovariant calculi $\Omega(A),\Omega(H)$ (or a braided version $\Omega(B)$). We use super versions of each of the constructions. Moreover, the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable. In the case of the Drinfeld double $D(A,H)=A^{\rm op}\bowtie H$ (where $A$ is dually paired to $H$), we show that its canonical actions on $A,H$ are differentiable. Examples include are a canonical $\Omega(\Bbb C_q[GL_2\ltimes \Bbb C^2])$ for the quantum group of affine transformations of the quantum plane and $\Omega(\Bbb C_\lambda[{\rm Poinc_{1,1}}])$ for the bicrossproduct Poincar\'e quantum group in 2 dimensions. We also show that $\Omega(\Bbb C_q[GL_2])$ itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra.

math.QA↗