Category Of Yetter-Drinfeld Modules Research Articles

Finite-dimensional Nichols algebras over the Suzuki algebras I: simple Yetter-Drinfeld modules of $A_{N\,2n}^{\mu\lambda}$

The Suzuki algebra $A_{Nn}^{\mu \lambda}$, introduced by Suzuki Satoshi in 1998, is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{\mu\lambda}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;\I_2)$ and the Nichols algebra $\ufo(8)$. There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N\,2n}^{\mu \lambda}$. The $64$-dimensional Nichols algebras are of dihedral rack type $\Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $\mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{\mu \lambda}$. Using a result of Masuoka, we prove that $\dim\mathfrak{B}(V_{abe})=\infty$ under the condition $b^2=(ae)^{-1}$, $b\in\Bbb{G}_{m}$ for $m\geq 5$.

The character algebra formodule categories over Hopf algebras

Given a finite dimensional Hopf algebra H and an exact indecomposable module category M over Rep(H), we explicitly compute the adjoint algebra A_M as an object in the category of Yetter-Drinfeld modules over H, and the space of class functions CF(M) associated to M, as introduced by K. Shimizu [ Further results on the structure of (Co)ends in fintite tensor categories, preprint arXiv:1801.02493]. We use our construction to describe these algebras when H is a group algebra and a dual group algebra. This result allows us to compute the adjoint algebra for certain group-theoretical fusion categories.

Non-degeneracy conditions for braided finite tensor categories

For a braided finite tensor category C with unit object 1∈C, Lyubashenko considered a certain Hopf algebra F∈C endowed with a Hopf pairing ω:F⊗F→1 to define the notion of a ‘non-semisimple’ modular tensor category. We say that C is non-degenerate if the Hopf pairing ω is non-degenerate. In this paper, we show that C is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its Müger center is trivial, if and only if the linear map HomC(1,F)→HomC(F,1) induced by the pairing ω is invertible. As an application, we prove that the category of Yetter-Drinfeld modules over a Hopf algebra in C is non-degenerate if and only if C is.

Cohomology and Coquasi-bialgebras in the category of Yetter-Drinfeld modules

We prove that a finite-dimensional Hopf algebra with the dual Chevalley Property over a field of characteristic zero is quasi-isomorphic to a RadfordMajid bosonization whenever the third Hochschild cohomology group in the category of Yetter-Drinfeld modules of its diagram with coefficients in the base field vanishes. Moreover we show that this vanishing occurs in meaningful examples where the diagram is a Nichols algebra.

Hopf Modules in the Category of Yetter-Drinfeld Modules

We give the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules , where L is a quasitriangular weak Hopf algebra with a bijective antipode. We also show that H* has a right H-Hopf module structure in the Yetter-Drinfeld category. As an application we deduce the existence and uniqueness of right integral from it.

Yetter-Drinfeld modules over weak braided Hopf monoids and center categories

In this paper, we introduce several centralizer constructions in a monoidal context and establish a monoidal equivalence with the category of Yetter-Drinfeld modules over a weak braided Hopf monoid. We apply the general result to the calculus of the center in module categories.

Quantum Cocommutative Coalgebras in $^{H}_{H}\mathcal{YD}$ and the Solutions of the Quasi-Yang-Baxter Equation

In this paper, we construct a quantum cocommutative coalgebra in the category of Yetter-Drinfeld modules over a coquasi-Hopf algebra, and give some solutions of the quasi-Yang-Baxter equation.

From Quantum Quasi-Shuffle Algebras to Braided Rota–Baxter Algebras

In this letter, we use quantum quasi-shuffle algebras to construct Rota-Baxter algebras, as well as tridendriform algebras. We also propose the notion of braided Rota-Baxter algebras, which is the relevant object of Rota-Baxter algebras in a braided tensor category. Examples of such new algebras are provided by using quantum multi-brace algebras in a category of Yetter-Drinfeld modules.

Braided Hopf Algebras Obtained from Coquasitriangular Hopf Algebras

Let $(H, \sigma)$ be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras, we define $H_\sigma$, a sub-Hopf algebra of $H^0$, the finite dual of $H$. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we associate to $H$ a braided Hopf algebra structure in the category of Yetter-Drinfeld modules over $H_\sigma^{\rm cop}$. Specializing to $H={\rm SL}_q(N)$, we obtain explicit formulas which endow ${\rm SL}_q(N)$ with a braided Hopf algebra structure within the category of left Yetter-Drinfeld modules over $U_q^{\rm ext}({\rm sl}_N)^{\rm cop}$.

Yetter-Drinfeld Categories for Quasi-Hopf Algebras

ABSTRACT We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra H are isomorphic. We prove also that the category of finite dimensional left Yetter-Drinfeld modules is rigid, and then we compute explicitly the canonical isomorphisms in . Finally, we show that certain duals of H 0, the braided Hopf algebra (introduced in Bulacu and Nauwelaerts, 2002; Bulacu et al., 2000) are isomorphic as braided Hopf algebras if H is a finite dimensional triangular quasi-Hopf algebra. Communicated by M. Takeuchi.

Yetter-Drinfeld modules over weak bialgebras

We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. IfH is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show that the category of Yetter-Drinfeld modules is isomorphic to the category of modules over the Drinfeld double. The category of finitely generated projective Yetter-Drinfeld modules over a weak Hopf algebra has duality.

Actions of Hopf Algebras in Categories of Yetter-Drinfeld Modules

Let H be a k-Hopf algebra and A a Hopf algebra in the category of Yetter-Drinfeld modules over H. We define the notion of an action of A on an H-module algebra and study inner and outer actions. Especially we prove that for each action of A there is a largest inner Hopf subalgebra in if we assume pointedness. Furthermore we prove the existence of non-trivial invariants of inner or outer actions of A. This gives new information about actions of biproducts.

BRAIDED MONOIDAL CATEGORIES ASSOCIATED TO YETTER-DRINFELD CATEGORIES

ABSTRACT Let be a Hopf algebra with invertible antipode, a Yetter-Drinfeld module algebra and the category of Hopf Yetter-Drinfeld -module. In this note it is discussed when the category is braided monoidal. In fact the following is proved: assume that is -commutative. Then the braiding on the category of Yetter-Drinfeld modules induces a braiding on if and only if every object of is dyslectic.

On Lie Algebras in the Category of Yetter-Drinfeld Modules

The category of Yetter—Drinfeld modules \(YD_K^K \) over a Hopf algebra K (with bijective antipode over a field k) is a braided monoidal category. If H is a Hopf algebra in this category then the primitive elements of H do not form an ordinary Lie algebra anymore. We introduce the notion of a (generalized) Lie algebra in \(YD_K^K \) such that the set of primitive elements P(H) is a Lie algebra in this sense. Also the Yetter—Drinfeld module of derivations of an algebra A in \(YD_K^K \) is a Lie algebra. Furthermore for each Lie algebra in \(YD_K^K \) there is a universal enveloping algebra which turns out to be a Hopf algebra in \(YD_K^K \).