The prime spectrum of the Drinfeld double of the Jordan plane

On the finite-dimensional representations of the double of the Jordan plane

We continue the study of the Drinfeld double of the Jordan plane, denoted by $\mathcal D$ and introduced in arXiv:2002.02514. The simple finite-dimensional modules were computed in arXiv:2108.13849; it turns out that they factorize through $U(\spl_2(\Bbbk))$. Here we introduce the Verma modules and the category $\mathfrak O$ for $\mathcal D$, which have a resemblance to the similar ones in Lie theory but induced from indecomposable modules of the 0-part of the triangular decomposition. Accordingly, there is the notion of highest weight rank (hw-rk). We classify the indecomposable modules of hw…

Modular quantizations of Lie algebras of Cartan type $H$ via Drinfeld Twists

We construct explicit Drinfel'd twists for the generalized Cartan type $H$ Lie algebras in characteristic $0$ and obtain the corresponding quantizations and their integral forms. Via making modular reductions including modulo $p$ reduction and modulo $p$-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{H}(2n;\underline{1}))$ in characteristic $p$. They are new non-pointed Hopf algebras of truncated $p$-polynomial noncommutative and noncocommutative deformation of prime-power dimension $p^{p^{2n…

Generalized δ-Derivations

We defined generalized δ-derivations of algebra A as linear mapping χassociated with usual δ-derivation ϕby the rule χ(xy)=δ(χ(x)y+xϕ(y))=δ(ϕ(x)y+xχ(y)) for any x,y \in A. We described generalized δ-derivations of prime alternative algebras, prime Lie algebras and superalgebras, unital algebras, and semisimple finite-dimensional Jordan superalgebras. In this cases we proved that generalized δ-derivation is a generalized derivation or δ-derivation. After that we described δ-superderivations of superalgebras <<KKM Double>>, arising from prime alternative algebras, prime Lie algebras and superalg…

Some quasitensor autoequivalences of Drinfeld doubles of finite groups

We report on two classes of autoequivalences of the category of Yetter-Drinfeld modules over a finite group, or, equivalently the Drinfeld center of the category of representations of a finite group. Both operations are related to the $r$-th power operation, with $r$ relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category defined by a finite group and an arbitrary subgroup, while the other seems particular to the case of Yetter-Drinfeld modules. Both autoequivalences preserve higher Frobenius-Schur indicators up to Galois conjugati…

Finite quantum groups over abelian groups of prime exponent

Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory. One of our main results is the explicit construction of a general family of pointed Hopf algebras from Dynkin diagrams. All the Frobenius-Lusztig kernels and their parabolic subalgebras belong to this family, but in addition we get many new ex…

Structure of semisimple Hopf algebras of dimension $p^2q^2$

Let $p,q$ be prime numbers with $p^4<q$, and $k$ an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension $p^2q^2$ can be constructed either from group algebras and their duals by means of extensions, or from Radford biproduct $R#kG$, where $kG$ is the group algebra of group $G$ of order $p^2$, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$. As an application, the special case that the structure of semisimple Hopf algebras of dimension $4q^2$ is given.

Braided Hopf algebras and gauge transformations

We study infinitesimal gauge transformations of an equivariant noncommutative principal bundle as a braided Lie algebra of derivations. For this, we analyse general $K$-braided Hopf and Lie algebras, for $K$ a (quasi)triangular Hopf algebra of symmetries, and study their representations as braided derivations. We then study Drinfeld twist deformations of braided Hopf algebras and of Lie algebras of infinitesimal gauge transformations. We give examples coming from deformations of abelian and Jordanian type. In particular we explicitly describe the braided Lie algebra of gauge transformations of…

高阶 Kirillov-Reshetikhin 模、虚模与 $U_q(A_n^{(1)})$ 的幺半范畴化

Higher order Kirillov-Reshetikhin modules, Imaginary modules and Monoidal Categorification for $U_q(A_n^{(1)})$

研究量子仿射李代数不可约模族，识别素模并证明唯一分解，推广范畴并定义膨胀，给出虚模存在性及显式公式，并应用于簇变量和 $D_4$ 型虚模。

Some Properties of Finite-Dimensional Semisimple Hopf Algebras

Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, th…

On Nichols algebras of low dimension

This is a contribution to the classification program of pointed Hopf algebras. We give a generalization of the quantum Serre relations and propose a generalization of the Frobenius-Lusztig kernels in order to compute Nichols algebras of diagonal group type. With this, we classify Nichols algebras B(V) with dimension < 32 or with dimension p^3, p a prime number, when V lies in a Yetter-Drinfeld category over a finite group. With the so called Lifting Procedure, this allows to classify pointed Hopf algebras of index < 32 or p^3. In recent articles by Etingof-Schedler-Soloviev and Lu-Yan-Zhu, the…