arXiv subjects

Van C. Nguyen

Publications and source records attributed to Van C. Nguyen.

At least 19 recordsLinked to original sources

arXiv2025

Quantum-symmetric equivalence for superpotential algebras

We study superpotential algebras by introducing the notion of quantum-symmetric equivalence defined relatively to two fixed Hopf coactions. This concept relies on the non-vanishing of a bi-Galois object for the two coacting Hopf algebras, where the cotensor product with this object provides a Morita--Takeuchi equivalence between their comodule categories, mapping one superpotenial algebra to the other as comodule algebras. In particular, we investigate $\mathcal{GL}$-type and $\mathcal{SL}$-type quantum-symmetric equivalences using Bichon's reformation of bi-Galois objects in the language of cogroupoids constructed by nondegenerate twisted superpotentials. As applications, for the $\mathcal{GL}$-type, we characterize the Artin--Schelter regularity, or equivalently, twisted Calabi--Yau property, of a superpotential algebra as the non-vanishing of the bi-Galois object in the associated cogroupoid. For the $\mathcal{SL}$-type, we apply the pivotal structure of the comodule categories to study numerical invariants for $\mathcal{SL}$ quantum-symmetric equivalence, including the quantum Hilbert series of the superpotential algebras.

math.QA↗

arXiv2025

On a minimal free resolution of the residue field over a local ring of codepth 3 of class $T$

Let $R$ be any noetherian local ring with residue field $k$, and $A$ the homology of the Koszul complex on a minimal set of generators of the maximal ideal of $R$. In this paper, we show that a minimal free resolution of $k$ over $R$ can be obtained from a graded minimal free resolution of $k$ over $A$. More precisely, this is done by the iterated mapping cone construction, introduced by the authors in a previous work, using specific choices of ingredients. As applications, using this general perspective, we exhibit a minimal free resolution of $k$ over a complete intersection ring of any codepth, and explicitly construct a minimal free resolution of $k$ over a noetherian local ring of codepth 3 of class $T$ in terms of Koszul blocks.

math.AC↗

arXiv2025

On the finite generation of the cohomology of bosonizations

We use deformation sequences of (Hopf) algebras, extending the results of Negron and Pevtsova, to show that bosonizations of some suitable braided Hopf algebras by some suitable finite-dimensional Hopf algebras have finitely generated cohomology. In fact, our results are shown in more generality for smash products. As applications, we prove the bosonizations of some Nichols algebras (such as Nichols algebras of diagonal type, the restricted Jordan plane, Nichols algebras of direct sums of Jordan blocks plus points labeled with 1), by some suitable finite-dimensional Hopf algebras, have finitely generated cohomology, recovering some known results as well as providing new examples.

math.QA↗

arXiv2024

Quantum-symmetric equivalence is a graded Morita invariant

We show that if two $m$-homogeneous algebras have Morita equivalent graded module categories, then they are quantum-symmetrically equivalent, that is, there is a monoidal equivalence between the categories of comodules for their associated universal quantum groups (in the sense of Manin) which sends one algebra to the other. As a consequence, any Zhang twist of an $m$-homogeneous algebra is a 2-cocycle twist by some 2-cocycle from its Manin's universal quantum group.

math.QA↗

arXiv2024

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Let $(R,\mathfrak m, \mathsf k)$ be a complete intersection local ring, $K$ be the Koszul complex on a minimal set of generators of $\mathfrak m$, and $A=H(K)$ be its homology algebra. We establish exact sequences involving direct sums of the components of $A$ and express the images of the maps of these sequences as homologies of iterated mapping cones built on $K$. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field $\mathsf k$ over $R$, independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns.

math.AC↗

arXiv2024

Twisting Manin's universal quantum groups and comodule algebras

We introduce the notion of quantum-symmetric equivalence of two connected graded algebras, based on Morita-Takeuchi equivalences of their universal quantum groups, in the sense of Manin. We study homological and algebraic invariants of quantum-symmetric equivalence classes, and prove that numerical $\mathrm{Tor}$-regularity, Castelnuovo-Mumford regularity, Artin-Schelter regularity, and the Frobenius property are invariant under any Morita-Takeuchi equivalence. In particular, by combining our results with the work of Raedschelders and Van den Bergh, we prove that Koszul Artin-Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class. Moreover, we characterize 2-cocycle twists (which arise as a special case of quantum-symmetric equivalence) of Koszul duals, of superpotentials, of superpotential algebras, of Nakayama automorphisms of twisted Frobenius algebras, and of Artin-Schelter regular algebras. We also show that finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists.

math.QA↗

arXiv2022

A cogroupoid associated to preregular forms

We construct a family of cogroupoids associated to preregular forms and recover the Morita-Takeuchi equivalence for Artin-Schelter regular algebras of dimension two, observed by Raedschelders and Van den Bergh. Moreover, we study the 2-cocycle twists of pivotal analogues of these cogroupoids, by developing a categorical description of preregularity in any tensor category that has a pivotal structure.

math.RA↗

arXiv2022

Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation

Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the Zhang twist of $H$ by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated to quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.

math.RA↗

arXiv2022

Tensor Representations for the Drinfeld Double of the Taft Algebra

Over an algebraically closed field $\mathbb k$ of characteristic zero, the Drinfeld double $D_n$ of the Taft algebra that is defined using a primitive $n$th root of unity $q \in \mathbb k$ for $n \geq 2$ is a quasitriangular Hopf algebra. Kauffman and Radford have shown that $D_n$ has a ribbon element if and only if $n$ is odd, and the ribbon element is unique; however there has been no explicit description of this element. In this work, we determine the ribbon element of $D_n$ explicitly. For any $n \geq 2$, we use the R-matrix of $D_n$ to construct an action of the Temperley-Lieb algebra $\mathsf{TL}_k(ξ)$ with $ξ= -(q^{\frac{1}{2}}+q^{-\frac{1}{2}})$ on the $k$-fold tensor power $V^{\otimes k}$ of any two-dimensional simple $D_n$-module $V$. This action is known to be faithful for arbitrary $k \geq 1$. We show that $\mathsf{TL}_k(ξ)$ is isomorphic to the centralizer algebra $\text{End}_{D_n}(V^{\otimes k})$ for $1 \le k \le 2n-2$.

math.RA↗

arXiv2021

New approaches to finite generation of cohomology rings

In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A$ to an algebraic variety using the cohomology ring of $A$. An essential assumption in this theory is the finite generation condition for the cohomology ring of $A$ and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra $A$, we show that the finite generation condition on $A$-modules can be replaced by a condition on any affine commutative $A$-module algebra $R$ under the assumption that $R$ is integral over its invariant subring $R^A$. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if $A$ is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra $A'$ over a finite field, where $A$ can be viewed as a deformation of $A'$, and prove that if the finite generation condition holds for $A'$, then the same condition holds for $A$.

math.RA↗

arXiv2021

A truncated minimal free resolution of the residue field

In a paper in 1962, Golod proved that the Betti sequence of the residue field of a local ring attains an upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components he also constructed a minimal free resolution of the residue field. In this article, we extend this construction up to degree five for any local ring. We describe how the multiplicative structure and the triple Massey products of the homology of the Koszul algebra are involved in this construction. As a consequence, we provide explicit formulas for the first six terms of a sequence that measures how far the ring is from being Golod.

math.AC↗

arXiv2021

McKay Matrices for Finite-dimensional Hopf Algebras

For a finite-dimensional Hopf algebra $A$, the McKay matrix $M_V$ of an $A$-module $V$ encodes the relations for tensoring the simple $A$-modules with $V$. We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $M_V$ by relating them to characters. We show how the projective McKay matrix $Q_V$ obtained by tensoring the projective indecomposable modules of $A$ with $V$ is related to the McKay matrix of the dual module of $V$. We illustrate these results for the Drinfeld double $D_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $M_V$ and $Q_V$ in terms of several kinds of Chebyshev polynomials. For the matrix $N_V$ that encodes the fusion rules for tensoring $V$ with a basis of projective indecomposable $D_n$-modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

math.RA↗

arXiv2018

The Gerstenhaber structure on the Hochschild cohomology of a class of special biserial algebras

We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0 and commute otherwise. Finally, we describe the cohomology in degree $n$ as a module over this Lie algebra by providing its decomposition as a direct sum of indecomposable modules.

math.RA↗

arXiv2017

Finite generation of the cohomology of some skew group algebras

We prove that some skew group algebras have Noetherian cohomology rings, a property inherited from their component parts. The proof is an adaptation of Evens' proof of finite generation of group cohomology. We apply the result to a series of examples of finite dimensional Hopf algebras in positive characteristic.

math.RT↗

arXiv2017

Finite generation of some cohomology rings via twisted tensor product and Anick resolutions

Over a field of prime characteristic $p>2$, we prove that the cohomology rings of some pointed Hopf algebras of dimension $p^3$ are finitely generated. These are Hopf algebras arising in the ongoing classification of finite dimensional Hopf algebras in positive characteristic, and include bosonizations of Nichols algebras of Jordan type in a general setting as well as their liftings when $p=3$. Our techniques are applications of twisted tensor product resolutions and Anick resolutions in combination with May spectral sequences.

math.RA↗

arXiv2017

Dominant dimension and tilting modules

We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension $2$, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least $2$, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting-cotilting module generated-cogenerated by projective-injective modules are precisely $1$-Auslander-Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

math.RT↗

arXiv2017

Preprojective algebras of tree-type quivers

Let $Q$ be a tree-type quiver, $\mathbf{k} Q$ its path algebra, and $λ$ a nonzero element in the field $\mathbf{k}$. We construct irreducible morphisms in the Auslander-Reiten quiver of the transjective component of the bounded derived category of $\mathbf{k} Q$ that satisfy what we call the $λ$-relations. When $λ=1$, the relations are known as mesh relations. When $λ=-1$, they are known as commutativity relations. Using this technique together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree-type quiver, several descriptions of its preprojective algebra are equivalent.

math.RA↗

arXiv2016

Pointed $p^3$-dimensional Hopf algebras in positive characteristic

We classify pointed $p^3$-dimensional Hopf algebras $H$ over any algebraically closed field $k$ of prime characteristic $p>0$. In particular, we focus on the cases when the group $G(H)$ of group-like elements is of order $p$ or $p^2$, that is, when $H$ is pointed but is not connected nor a group algebra. This work provides many new examples of (parametrized) non-commutative and non-cocommutative finite-dimensional Hopf algebras in positive characteristic.

math.RA↗