Related papers: The Drinfel'd codouble constuction for monoidal Ho…

For a quasi-Hopf algebra $H$, a left $H$-comodule algebra $\mf{B}$ and a right $H$-module coalgebra $C$ we will characterize the category of Doi-Hopf modules ${}^C{\cal M}(H)_{\mf{B}}$ in terms of modules. We will also show that for an…

We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the…

The aim of this paper is to define and study Yetter-Drinfeld modules over weak Hom-Hopf algebras. We show that the category ${}_H{\cal WYD}^H$ of Yetter-Drinfeld modules with bijective structure maps over weak Hom-Hopf algebras is a rigid…

The cyclic (co)homology of Hopf algebras is defined by Connes and Moscovici [math.DG/9806109] and later extended by Khalkhali et.al [math.KT/0306288] to admit stable anti-Yetter-Drinfeld coefficient module/comodules. In this paper we will…

In this article, we establish the duality between the generalised Drinfeld double and generalised quantum codouble within the framework of modular or manageable (not necessarily regular) multiplicative unitaries, and discuss several…

Let $(S,L)$ be a Lie-Rinehart algebra such that $L$ is $S$-projective and let $U$ be its universal enveloping algebra. In this paper we present a spectral sequence which converges to the Hochschild cohomology of $U$ with values on a…

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of…

We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf-Pasquier-Roche quasi-Hopf algebra D^{omega}(Sigma), for some finite group Sigma and some 3-cocycle omega on…

In our earlier article [Lett. Math. Phys. 107 (2017), 475-503, arXiv:1409.8188], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every…

We study versions of the categories of Yetter-Drinfel'd modules over a Hopf algebra $H$ in a braided monoidal category $\C$. Contrarywise to Bespalov's approach, all our structures live in $\C$. This forces $H$ to be transparent or…

We classify equivalence classes of Hopf algebra quotient pairs $(D,\theta)$ of the Drinfeld double $D(G)$ of a finite group scheme $G$ over an algebraically closed field $\mathbf{k}$ of characteristic $p\ge 0$, in terms of group…

In order to study the Hochschild cohomology of triangular algebras $\mathcal T$, we construct a spectral sequence, whose terms are parametrized by the length of the trajectories of the quiver associated with $\mathcal T$, and which…

We associate to a projective $n$-dimensional toric variety $X_{\Delta}$ a pair of co-commutative (but generally non-commutative) Hopf algebras $H^{\alpha}_X, H^{T}_X$. These arise as Hall algebras of certain categories $\Coh^{\alpha}(X),…

We define an equivariant $K_0$-theory for \textit{Yetter-Drinfeld} algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a…

Let $H$ be a finite dimensional quasi-Hopf algebra over a field $k$ and ${\mathfrak A}$ a right $H$-comodule algebra in the sense of Hausser and Nill. We first show that on the $k$-vector space ${\mathfrak A}\ot H^*$ we can define an…

We introduce a quantum double quasitriangular quasi-Hopf algebra $D(H)$ associated to any quasi-Hopf algebra $H$. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover…

We describe a general framework for studying duality between different phase spaces which share the same symmetry group $\mathrm{H}$. Solutions corresponding to collective dynamics become dual in the sense that they are generated by the…

An interesting open question is to determine the group of monoidal autoequivalences of the category of Yetter-Drinfeld modules over a finite group $G$, or equivalently the group of Bigalois objects over the dual of the Drinfeld double $DG$.…

Within the framework of braided or quasisymmetric monoidal categories braided Q-supersymmetry is investigated, where Q is a certain functorial isomorphism in a braided symmetric monoidal category. For an ordinary (co-)quasitriangular Hopf…

Generalising a result for Hopf algebras, we not only define the four possible types of Hopf modules in the bialgebroid setting but also yield the notion of two-sided two-cosided Hopf modules, also known as Hopf bimodules or tetramodules, in…