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Naihong Hu

Publications and source records attributed to Naihong Hu.

At least 37 records · Page 2Linked to original sources
arXiv2023

Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

Let $\mathbb{K}$ be a field. We study the free bialgebra $\mathcal{T}$ generated by the coalgebra $C=\mathbb{K} g \oplus \mathbb{K} h$ and its quotient bialgebras (or Hopf algebras) over $\mathbb{K}$. We show that the free noncommutative Faà di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$, and the quotient bialgebra $\overline{\mathcal{T}}:=\mathcal{T}/(E_α|~α(g)\ge 2)$ is an Ore extension of the well-known Faà di Bruno bialgebra. The image of the free noncommutative Faà di Bruno bialgebra in the quotient $\overline{\mathcal{T}}$ gives a more reasonable non-commutative version of the commutative Faà di Bruno bialgebra from the PBW basis point view. If char $\mathbb{K}=p>0$, we obtain a chain of quotient Hopf algebras of $\overline{\mathcal{T}}$: $\overline{\mathcal{T}} \twoheadrightarrow \overline{\mathcal{T}}_{n}\twoheadrightarrow \overline{\mathcal{T}}_{n}'(p)\twoheadrightarrow \overline{\mathcal{T}}_{n}(p)\twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{1}) \twoheadrightarrow \ldots \twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{j},d_{j-1},\ldots,d_{1}) \twoheadrightarrow \ldots \twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{p-2},d_{p-3},\ldots,d_{1})$ with finite GK-dimensions. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.

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arXiv2022

On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^*$ of Kashina

Let $H$ be the dual of $16$-dimensional nontrivial semisimple Hopf algebra $H_{b:1}$ in the classification work of Kashina \cite{K00}. We completely determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(N)\cong \bigotimes_{i\in I}\mathcal{B}(N_i)$, where $N=\bigoplus_{i\in I}N_i$, each $N_i$ is a simple object in $_H^H\mathcal{YD}$. Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra $H$ via the relevant Nichols algebras $\mathcal B(N)$.

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arXiv2022

Classification of finite-dimensional Hopf algebras over dual Radford algebras

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we obtain families of new examples of finite-dimensional Hopf algebras without the dual Chevalley property.

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arXiv2021

Drinfel'd doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial

In the paper, we describe the Drinfel'd double structure of the $n$-rank Taft algebra and all of its simple modules, and then endow its $R$-matrices with some application to knot invariants. The knot invariants we get is a generalization of the Jones polynomial, in particular, it recovers the Jones polynomial in rank $1$ case, while in rank $2$ case, it is the one-parameter specialization of the two-parameter unframed Dubrovnik polynomial, and in higher rank case it is the composite ($n$-power) of the Jones polynomial.

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arXiv2019

Finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}$ of Kashina

Let $H$ be the 16-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:1}$ appeared in Kashina's work \cite{K00}. We obtain all simple Yetter-Drinfeld modules over $H$ and then determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(V)\cong \bigotimes_{i\in I}\mathcal{B}(V_i)$, where $V=\bigoplus_{i\in I}V_i$, each $V_i$ is a simple object in $_H^H\mathcal{YD}$. Finally, we describe all liftings of those $\mathcal{B}(V)$ over $H$.

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arXiv2019

Quantum (dual) Grassmann superalgebra as $\mathcal U_q(\mathfrak{gl}(m|n))$-module algebra and beyond

We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $Ω_q(m|n)$. Correspondingly, a quantum Weyl algebra $\mathcal W_q(2(m|n))$ of $(m|n)$-type is introduced as the quantum differential operators (QDO for short) algebra $\textrm{Diff}_q(Ω_q)$ defined over $Ω_q(m|n)$, which is a smash product of the quantum differential Hopf algebra $\mathfrak D_q(m|n)$ (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra $Ω_q(m|n)$. An interested point of this approach here is that even though $\mathcal W_q(2(m|n))$ itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This point of view gives us one of main expected results, that is, the quantum (restricted) Grassmann superalgebra $Ω_q$ is made into the $\mathcal U_q(\mathfrak g)$-module (super)algebra structure,$Ω_q=Ω_q(m|n)$ for $q$ generic, or $Ω_q(m|n, \bold 1)$ for $q$ root of unity, and $\mathfrak g=\mathfrak{gl}(m|n)$ or $\mathfrak {sl}(m|n)$, the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple $\mathcal U_q(\mathfrak g)$-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra $Ω_q^!$ as $\mathcal U_q(\mathfrak g)$-module algebra.In the paper some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in \cite{Ma}, \& \cite{Ma1}.

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arXiv2017

Realization of $U_q({\mathfrak{sp}}_{2n})$ within the Differential Algebra on Quantum Symplectic Space

We realize the Hopf algebra $U_q({\mathfrak {sp}}_{2n})$ as an algebra of quantum differential operators on the quantum symplectic space $\mathcal{X}(f_s;\mathrm{R})$ and prove that $\mathcal{X}(f_s;\mathrm{R})$ is a $U_q({\mathfrak{sp}}_{2n})$-module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root vectors under the actions of Lusztig's braid automorphisms of $U_q({\mathfrak {sp}}_{2n})$.

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arXiv2017

Eight classes of new Hopf algebras of dimension $128$ without the Chevalley property

Classifying Hopf algebras of a given dimension is a hard and open question. Using the generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf algebra $H$ of dimension $16$ without the Chevalley property and the corresponding infinitesimal braidings are simple objects in $\HYD$. In particular, we figure out $8$ classes of new Hopf algebras of dimension $128$ without the Chevalley property.

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arXiv2016

Some Hopf algebras of dimension $72$ without the Chevalley property

In this paper, we consider the Drinfeld double $\D$ of a $12$-dimensional Hopf algebra $\C$ over an algebraically closed field of characteristic zero whose coradical is not a subalgebra and describe its simple modules, projective covers of the simple modules and show that it is of wild representation type. Moreover, we show that the Nichols algebras associated to non-simple indecomposable modules are infinite-dimensional. In particular, for any object $V$ in $\CYD$, if $\BN(V)$ is finite-dimensional, then $V$ must be semisimple. Finally, we describe the Nichols algebras associated to partial simple modules in terms of generators and relations. As a byproduct, we obtain some Hopf algebras of dimension $72$ without the Chevalley property, that is, the coradical is not a subalgebra.

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arXiv2016

Convex PBW-type Lyndon Bases and Restricted Two-Parameter Quantum Group of Type $F_4$

We determine convex PBW-type Lyndon bases for two-parameter quantum groups $U_{r,s}(F_4)$ with detailed commutation relations. We construct a finite-dimensional Hopf algebra $\mathfrak u_{r,s}(F_4)$, as a quotient of $U_{r,s}(F_4)$ by a Hopf ideal generated by certain central elements, which is pointed, and of a Drinfel'd double structure under a certain condition. All of Hopf isomorphisms of $\mathfrak u_{r,s}(F_4)$ are determined which are important for seeking the possible new pointed objects in low order with $(\ell, 210)\ne 1$. Finally, necessary and sufficient conditions for $\mathfrak u_{r,s}(F_4)$ to be a ribbon Hopf algebra are singled out by describing the left and right integrals.

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arXiv2016

Double-bosonization and Majid's Conjecture, (III): type-crossing and inductions of $E_6$ and $E_7$, $E_8$

Double-bosonization construction in Majid \cite{majid1} is expectedly allowed to generate a tree of quantum groups. Some main branches of the tree in \cite{HH1, HH2} have been depicted how to grow up. This paper continues to elucidate the type-crossing and inductive constructions of exceptional quantum groups of types $E_6$ and $E_7$, $E_8$, respectively, based on the generalized double-bosonization Theorem established in \cite{HH2}. Thus the Majid's expectation for the inductive constructions of $U_q(\mathfrak g)$'s for all finite-dimensional complex simple Lie algebras is completely achieved.

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arXiv2015

Double-bosonization and Majid's Conjecture, (II): cases of irregular $R$-matrices and type-crossings of $F_4$, $G_2$

The purpose of the paper is to build up the related theory of weakly quasitriangular dual pairs suitably for non-standard $R$-matrices (irregular), and establish the generalized double-bosonization construction theorem for irregular $R$, which generalize Majid's results for regular $R$ in \cite{majid1}. As an application, the type-crossing construction for the exceptional quantum groups of types $F_{4}$, $G_{2}$ is obtained. This affirms the Majid's expectation that the tree structure of nodes diagram associated with quantum groups can be grown out of the node corresponding to $U_q(\mathfrak{sl}_2)$ by double-bosonization procedures. Notably from a representation perspective, we find an effective method to get the minimal polynomials for the non-standard $R$-matrices involved.

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arXiv2015

Double-bosonization and Majid's Conjecture, (IV): Type-Crossings from $A$ to $BCD$

Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rank-inductive and type-crossing construction for $U_q(\mathfrak g)$'s is still a remaining open question. In this paper, working with Majid's framework, based on our generalized double-bosonization Theorem proved in \cite{HH2}, we further describe explicitly the type-crossing construction of $U_q(\mathfrak g)$'s for $(BCD)_n$ series direct from type $A_{n-1}$ via adding a pair of dual braided groups determined by a pair of $(R, R')$-matrices of type $A$ derived from the respective suitably chosen representations. %which generalize the lower rank cases constructed in \cite{HH1}. Combining with our work in \cite{HH1,HH2,HH3}, this solves Majid's conjecture, that is, any quantum group $U_q(\mathfrak g)$ associated to a simple Lie algebra $\mathfrak g$ can be grown out of $U_q({\mathfrak {sl}}_2)$ inductively by a series of suitably chosen double-bosonization procedures.

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arXiv2015

Modular Quantizations of Lie Algebras of Cartan Type K via Drinfeld Twists of Jordanian Type

We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{K}(2n{+}1;\underline{1}))$ for the restricted simple Lie algebra of Cartan type K in characteristic p. They are new pointed Hopf algebras of noncommutative and noncocommutative and with dimension $p^{p^{2n+1}+1}$ (if $2n+4\not\equiv0 \; (\mod p)$) or $p^{p^{2n+1}}$ (if $2n+4\equiv0 \; (\mod p))$ over a truncated p-polynomials ring, which also contain the well-known Radford algebras as Hopf subalgebras. Some open questions are proposed.

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arXiv2015

Double-bosonization and Majid's conjecture, (I): rank-induction of $ABCD$

Majid developed in \cite{majid3} his double-bosonization theory to construct $U_q(\mathfrak g)$ and expected to generate inductively not just a line but a tree of quantum groups starting from a node. In this paper, the authors confirm the Majid's first expectation (see p. 178 \cite{majid3}) through giving and verifying the full details of the inductive constructions of $U_q(\mathfrak g)$ for the classical types, i.e., the $ABCD$ series. Some examples in low ranks are given to elucidate that any quantum group of classical type can be constructed from the node corresponding to $U_{q}(\mathfrak{sl}_2)$.

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arXiv2015

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 $p^{p^{2n}-1}$, which contain the well-known Radford algebras as Hopf subalgebras. As a by-product, we also get some Jordanian quantizations for $\mathfrak {sp}_{2n}$.

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