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Yinhuo Zhang

Publications and source records attributed to Yinhuo Zhang.

At least 19 recordsLinked to original sources
arXiv2026

A Noetherian Hopf algebra is affine iff its Hopf coradical is affine

Let $\K$ denote a field. Extending the structural frameworks established in \cite{JZ2025-2}, this paper introduces novel techniques utilizing non-commutative reduction orders, factorization theory, and the generalized lifting methodology. We establish a definitive necessary and sufficient criterion for the affineness of Noetherian Hopf algebras, thereby providing a significant advancement toward resolving the long-standing Wu--Zhang question \cite{WZ2003}. Specifically, we prove that a left or right Noetherian Hopf algebra over $\K$ is affine if and only if its Hopf coradical is affine. This characterization fundamentally concentrates the burden of verification onto the first filtration step, yielding a criterion that is structurally transparent and highly operational. To establish necessity, we provide an essential intrinsic result demonstrating that the Hopf coradical of an affine Hopf algebra inherits the property of being affine. Furthermore, as direct applications of this equivalence, we prove that a left or right Noetherian Hopf algebra $H$ is affine provided that its coradical $H_{(0)}$ forms a subalgebra (the dual Chevalley property), its coradical $H_{(0)}$ is cocommutative, or its Hopf coradical $H_{[0]}$ is commutative.

math.RA↗
arXiv2026

Notes on gamma invariants of finite dimensional Hopf algebras

Let $H$ be a finite-dimensional, non-semisimple Hopf algebra over an algebraically closed field $\mathbf{k}$. This paper investigates the asymptotic behavior of the core of left $H$-modules through the lens of the gamma invariant $γ_{\mathfrak{X}}$ relative to a representation ideal $I_{\mathfrak{X}}$. We establish an equivalent characterization for the quotient of the Green ring $R_{\mathfrak{X}}$ to be a transitive fusion ring, demonstrating that transitivity is synonymous with the non-degeneracy of a naturally induced bilinear form and the collapse of the ideals $P_{+}$, $P_{-}$ and $I_{\operatorname{max}}$ into a single ideal. Furthermore, we prove that the Green ring exhibits the structure of a representation ring in the sense of Benson, provided that the square of the antipode is an inner automorphism and the equality $I_{\operatorname{max}}=I_{\operatorname{proj}}$ holds. As an explicit application of these frameworks, we analyze the Drinfeld double $D(H_4)$ of the Sweedler algebra, identifying an infinite family of distinct representation ideals and proving that the maximal gamma invariant $γ_{\operatorname{max}}$ induces a genuine ring homomorphism. Finally, for Hopf algebras of finite representation type under the assumption $P_{+} = P_{-} = I_{\operatorname{max}}$, we show that $γ_{\operatorname{max}}$ coincides precisely with the Frobenius--Perron dimension, and we explicitly compute the gamma invariants for the standard basis elements of the Green ring of the Taft algebra $H_n(q)$.

math.QA↗
arXiv2025

The Projective Class Rings of Drinfeld doubles of pointed rank one Hopf algebras

Let $\Bbbk$ be an algebraically closed field of characteristic $0$. In this paper, we study the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$ of the Drinfeld double $D(H_{\mathcal{D}})$ of the rank one pointed Hopf algebra $H_{\mathcal{D}}$. We analyze the tensor products of simple modules with simple modules, simple modules with indecomposable projective modules, and indecomposable projective modules with indecomposable projective modules, providing explicit decomposition rules in each case. Finally, we compute both the Grothendieck ring $G_0(D(H_\mathcal{D}))$ and the projective class ring $r_p(D(H_\mathcal{D}))$, and present these two rings in terms of generators and defining relations.

math.QA↗
arXiv2025

Noetherian pointed Hopf algebras are affine

Let $k$ be a field. In this paper, we introduce the notions of $\textit{reduction order}$ and $\textit{reduction-factorization}$ on words, and use them to show that any right or left Noetherian pointed Hopf algebra over $k$ is affine. This result offers a partial affirmative answer to the classical affineness question for Noetherian Hopf algebras posed by Wu and Zhang \cite{WZ2003}. For a pointed Hopf algebra $H$ over $k$, we construct a well-ordered set $X$ such that: (1) $H$ is generated, as an algebra, by the subset $X_I$ of irreducible letters (with respect to the reduction order); and (2) $X_I$ is finite whenever $H$ is right Noetherian.

math.QA↗
arXiv2025

Representations of the Drinfeld doubles of Pointed rank one Hopf algebras

In this paper, we investigate the representations of the Drinfeld doubles $D(H_{\mathcal{D}})$ of pointed rank one Hopf algebras $H_{\mathcal{D}}$ over an algebraically closed field $\Bbbk$ of characteristic zero. We provide a complete classification of all finite-dimensional indecomposable $D(H_{\mathcal{D}})$-modules up to isomorphism and explicitly describe the Auslander-Reiten sequences in the category of finite-dimensional $D(H_{\mathcal{D}})$-modules. We show that $D(H_{\mathcal{D}})$ is of tame representation type.

math.QA↗
arXiv2025

Affineness on Noetherian graded rings, algebras and Hopf algebras

In this note, we show that every Noetherian graded ring with an affine degree zero part is affine. As a result, a Noetherian graded Hopf algebra whose degree zero component is a commutative or a cocommutative Hopf subalgebra is affine. Moreover, we show that the braided Hopf algebra of a Noetherian graded Hopf algebra is affine.

math.RA↗
arXiv2023

Representations of the small quasi-quantum group

In this paper, we study the representation theory of the small quantum group $\overline{U}_q$ and the small quasi-quantum group $\widetilde{U}_q$, where $q$ is a primitive $n$-th root of unity and $n>2$ is odd. All finite dimensional indecomposable $\widetilde{U}_q$-modules are described and classified. Moreover, the decomposition rules for the tensor products of $\widetilde{U}_q$-modules are given. Finally, we describe the structures of the projective class ring $r_p(\widetilde{U}_q)$ and the Green ring $r(\widetilde{U}_q)$. We show that $r(\overline{U}_q)$ is isomorphic to a subring of $r(\widetilde{U}_q)$, and the stable Green rings $r_{st}(\widetilde{U}_q)$ and $r_{st}(\overline{U}_q)$ are isomorphic.

math.QA↗
arXiv2023

Noncommutative binomial theorem, shuffle type polynomials and Bell polynomials

In this paper we use the Lyndon-Shirshov basis to study the shuffle type polynomials. We give a free noncommutative binomial (or multinomial) theorem in terms of the Lyndon-Shirshov basis. Another noncommutative binomial theorem given by the shuffle type polynomials with respect to an adjoint derivation is established. As a result, the Bell differential polynomials and the $q$-Bell differential polynomials can be derived from the second binomial theorem. The relation between the shuffle type polynomials and the Bell differential polynomials is established. Finally, we give some applications of the free noncommutative binomial theorem including application of the shuffle type polynomials to bialgebras and Hopf algebras.

math.CO↗
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.

math.QA↗
arXiv2021

Representations of Hopf-Ore extensions of group algebras

In this paper, we study the representations of the Hopf-Ore extensions $kG(χ^{-1}, a, 0)$ of group algebra $kG$, where $k$ is an algebraically closed field. We classify all finite dimensional simple $kG(χ^{-1}, a, 0)$-modules under the assumption $|χ|=\infty$ and $|χ|=|χ(a)|<\infty$ respectively, and all finite dimensional indecomposable $kG(χ^{-1}, a, 0)$-modules under the assumption that $kG$ is finite dimensional and semisimple, and $|χ|=|χ(a)|$. Moreover, we investigate the decomposition rules for the tensor product modules over $kG(χ^{-1}, a, 0)$ when char$(k)$=0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra $kD_n$, where $n=2m$, $m>1$ odd, and char$(k)$=0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.

math.RA↗
arXiv2018

Reconstruction of tensor categories from their structure invariants

In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $\mathbb F$. Given a tensor category $\mathcal{C}$, we have two structure invariants of $\mathcal{C}$: the Green ring (or the representation ring) $r(\mathcal{C})$ and the Auslander algebra $A(\mathcal{C})$ of $\mathcal{C}$. We show that a Krull-Schmit abelian tensor category $\mathcal{C}$ of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of $\mathcal{C}$. In fact, we can reconstruct the tensor category $\mathcal{C}$ from its two invarinats and the associator system. More general, given a quadruple $(R, A, ϕ, a)$ satisfying certain conditions, where $R$ is a $\mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $\mathbb F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $ϕ$ is an algebra map from $A\otimes_{\mathbb F}A$ to an algebra $M(R, A, n)$ constructed from $A$ and $R$, and $a=\{a_{i,j,l}|1< i,j,l<n\}$ is a family of "invertible" matrices over $A$, we can construct a Krull-Schmidt and abelian tensor category $\mathcal C$ over $\mathbb{F}$ such that $R$ is the Green ring of $\mathcal C$ and $A$ is the Auslander algebra of $\mathcal C$. In this case, $\mathcal C$ has finitely many indecomposable objects (up to isomorphisms) and finite dimensional Hom-spaces. Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.

math.CT↗
arXiv2017

Local cohomology associated to the radical of a group action on a noetherian algebra

An arbitrary group action on an algebra $R$ results in an ideal $\mathfrak{r}$ of $R$. This ideal $\mathfrak{r}$ fits into the classical radical theory, and will be called the radical of the group action. If $R$ is a noetherian algebra with finite GK-dimension and $G$ is a finite group, then the difference between the GK-dimensionsof $R$ and that of $R/\mathfrak{r}$ is called the pertinency of the group action. We provide some methods to find elements of the radical, which helps to calculate the pertinency of some special group actions. The $\mathfrak{r}$-adic local cohomology of $R$ is related to the singularities of the invariant subalgebra $R^G$. We establish an equivalence between the quotient category of the invariant $R^G$ and that of the skew group ring $R*G$ through the torsion theory associated to the radical $\mathfrak{r}$. With the help of the equivalence, we show that the invariant subalgebra $R^G$ will inherit certain Cohen-Macaulay property from $R$.

math.RA↗
arXiv2017

Cohen-Macaulay invariant subalgebras of Hopf dense Galois extensions

Let $H$ be a semisimple Hopf algebra, and let $R$ be a noetherian left $H$-module algebra. If $R/R^H$ is a right $H^*$-dense Galois extension, then the invariant subalgebra $R^H$ will inherit the AS-Cohen-Macaulay property from $R$ under some mild conditions, and $R$, when viewed as a right $R^H$-module, is a Cohen-Macaulay module. In particular, we show that if $R$ is a noetherian complete semilocal algebra which is AS-regular of global dimension 2 and $H=\operatorname{\bf k} G$ for some finite subgroup $G\subseteq Aut(R)$, then all the indecomposable Cohen-Macaulay module of $R^H$ is a direct summand of $R_{R^H}$, and hence $R^H$ is Cohen-Macaulay-finite, which generalizes a classical result for commutative rings. The main tool used in the paper is the extension groups of objects in the corresponding quotient categories.

math.RA↗
arXiv2017

On nondiagonal finite quasi-qantum groups over finite abelian groups

In this paper, we initiate the study of nondiagonal finite quasi-quantum groups over finite abelian groups. We mainly study the Nichols algebras in the twisted Yetter-Drinfeld module category $_{\k G}^{\k G}\mathcal{YD}^Φ$ with $Φ$ a nonabelian $3$-cocycle on a finite abelian group $G.$ A complete clarification is obtained for the Nichols algebra $B(V)$ in case $V$ is a simple twisted Yetter-Drinfeld module of nondiagonal type. This is also applied to provide a complete classification of finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups of odd order and confirm partially the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik.

math.QA↗
arXiv2017

Finite-dimensional quasi-Hopf algebras of Cartan type

In this paper, we present a general method for constructing finite-dimensional quasi-Hopf algebras from finite abelian groups and braided vector spaces of Cartan type. The study of such quasi-Hopf algebras leads to the classification of finite-dimensional radically graded basic quasi-Hopf algebras over abelian groups with dimensions not divisible by $2,3,5,7$ and associators given by abelian $3$-cocycles. As special cases , the small quasi-quantum groups are introduced and studied. Many new explicit examples of finite-dimensional genuine quasi-Hopf algebras are obtained.

math.QA↗
arXiv2017

On the center of the quantized enveloping algebra of a simple Lie algebra

Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the quantum group $U_q(\frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(\frak{g}))$ is a polynomial algebra if and only if $\frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $\frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ variables with one relation; in case $\frak{g}$ is of type $E_6$, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in fourteen variables with eight relations; in case $\frak{g}$ is of type $A_{n}$, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra described by $n$-sequences.

math.QA↗