arXiv subjects

Ayako Itaba

Publications and source records attributed to Ayako Itaba.

11 recordsLinked to original sources

arXiv2026

Braided cohomology of quasi-triangular bialgebras and braided Morita invariance

We introduce the braided cochain complex and the braided cohomology of braided coalgebras in linear monoidal categories, and compare the braided cohomology of braided coalgebras living in different linear monoidal categories using relative morphisms. The symmetric cohomology was introduced for groups by Staic, and was generalized to cocommutative Hopf algebras by Shiba, Sanada, and the second author. This cohomology involves degreewise actions of the symmetric groups on a cochain complex, which come from the usual symmetric monoidal structure on the category of modules. We generalize this framework by dealing with arbitrary linear monoidal categories, and by replacing symmetries with braidings defined merely on an object. We first give a convenient description of relative morphisms, and apply this result to prove that the braided cochain complex of quasi-triangular bialgebras is a braided Morita invariant under a certain condition, which is automatically satisfied in the finite-dimensional case.

math.QA↗

arXiv2025

Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$)

Let $A=A(\alpha, \beta)$ be a graded down-up algebra with weights $(\mathrm{deg}\, x, \mathrm{deg}\, y)=(n,m)$ and $\beta \neq 0$, and $\nabla A$ its Beilinson algebra. Such an algebra $A$ is a 3-dimensional cubic AS-regular algebra by Kirkman--Musson--Passman. Assuming $\mathrm{gcd}\,(n, m)=1$ and $m \geq n$, we extend the previous results on the Hochschild cohomology of $\nabla A$. Known cases include $(n,m) = (1,1)$ (Belmans) and $(n = 1,\,m \geq 2)$ (Itaba--Ueyama). In this paper, we determine the dimensions of the Hochschild cohomology groups of $\nabla A$ in the remaining case $n\geq 2$ and $m\geq 2$ by explicitly constructing the projective resolution and computing the ranks of the arising representation matrices. As a byproduct, for $m>n>1$, we show that the derived category of the noncommutative projective scheme associated to $A$ is not equivalent to the derived category of any smooth projective surface. Moreover, for all $m \geq n \geq 1$, we describe the ring structure of the Hochschild cohomology group $\nabla A$ with respect to the Yoneda product.

math.RA↗

arXiv2025

Classifications of 3-dimensional cubic AS-regular algebras whose point schemes are not integral

By the result of Artin--Tate--Van den Bergh, every $3$-dimensional cubic AS-regular algebra A can be expressed as a geometric algebra $A=\mathcal{A}(E,\sigma)$, where $E$ is either $\mathbb{P}^{1}\times \mathbb{P}^{1}$ or a curve of bidegree ($2$,$2$) in $\mathbb{P}^{1}\times \mathbb{P}^{1}$ and $\sigma\in \mathrm{Aut}_{k}E$. In particular, we treat the following three configurations: (1) a conic and two lines in a triangle, (2) a conic and two lines intersecting in one point, and (3) a quadrangle. For each of these cases, we (i) list all defining relations of the corresponding algebras $\mathcal{A}(E,\sigma)$, and (ii) classify them up to graded algebra isomorphism and graded Morita equivalence. Furthermore, we present explicit (twisted) superpotentials whose derivation-quotient algebras realize these algebras and verify that the resulting algebras are AS-regular. Combining our results with existing classifications for the remaining types (including Types P, S, T, WL, and TWL), we thereby complete the classification of 3-dimensional cubic AS-regular algebras whose point schemes are not integral.

math.RA↗

arXiv2023

Quantum projective planes and Beilinson algebras of $3$-dimensional quantum polynomial algebras for Type S'

Let $A=\mathcal{A}(E,σ)$ be a $3$-dimensional quantum polynomial algebra where $E$ is $\mathbb{P}^{2}$ or a cubic divisor in $\mathbb{P}^{2}$, and $σ\in \mathrm{Aut}_{k}E$. Artin-Tate-Van den Bergh proved that $A$ is finite over its center if and only if the order $|σ|$ of $σ$ is finite. As a categorical analogy of their result, the author and Mori showed that the following conditions are equivalent; (1) $|ν^{\ast}σ^{3}|<\infty$, where $ν$ is the Nakayama automorphism of $A$. (2) The norm $\|σ\|$ of $σ$ is finite. (3) The quantum projective plane $\mathsf{Proj}_{\rm nc}A$ is finite over its center. In this paper, we will prove for Type S' algebra $A$ that the following conditions are equivalent; (1) $\mathsf{Proj}_{\rm nc}A$ is finite over its center. (2) The Beilinson algebra $\nabla A$ of $A$ is $2$-representation tame. (3) The isomorphism classes of simple $2$-regular modules over $\nabla A$ are parametrized by $\mathbb{P}^{2}$.

math.RA↗

arXiv2022

Symmetric cohomology and symmetric Hochschild cohomology of cocommutative Hopf algebras

Staic defined symmetric cohomology of groups and studied that the secondary symmetric cohomology group is corresponding to group extensions and the injectivity of the canonical map from symmetric cohomology to classical cohomology. In this paper, we define symmetric cohomology and symmetric Hochschild cohomology for cocommutative Hopf algebras. The first one is a generalization of symmetric cohomology of groups. We give an isomorphism between symmetric cohomology and symmetric Hochschild cohomology, which is a symmetric version of the classical result about cohomology of groups by Eilenberg-MacLane and cohomology of Hopf algebras by Ginzburg-Kumar. Moreover, to consider the condition that symmetric cohomology coincides with classical cohomology, we investigate the projectivity of a resolution which gives symmetric cohomology.

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arXiv2021

AS-regularity of geometric algebras of plane cubic curves

Let $k$ be an algebraically closed field of characteristic $0$ and $A$ a graded $k$-algebra finitely generated in degree $1$. In this paper, for $3$-dimensional quadratic AS-regular algebras except for Type EC, we give a complete list of twisted superpotentials and a complete list of superpotentials such that derivation-quotient algebras are $3$-dimensional quadratic Calabi-Yau AS-regular algebras. For an algebra $A$ of Type EC, we give a criterion when $A$ is AS-regular. As an application, for an algebra $A$ of any type, we show that there exists a Calabi-Yau AS-regular algebra $S$ such that $A$ and $S$ are graded Morita equivalent. This result tells us that, for a $3$-dimensional quadratic AS-regular algebra $A$, to study the noncommutative projective scheme for $A$ defined by Artin-Zhang is reduced to study the noncommutative projective scheme for $S$ for the Calabi-Yau AS-regular algebra $S$.

math.RA↗

arXiv2020

Quantum Projective Planes Finite over their Centers

For a $3$-dimensional quantum polynomial algebra $A=\mathcal{A}(E,σ)$, Artin-Tate-Van den Bergh showed that $A$ is finite over its center if and only if $|σ|<\infty$. Moreover, Artin showed that if $A$ is finite over its center and $E\neq \mathbb{P}^2$, then $A$ has a fat point module, which plays an important role in noncommutative algebraic geometry, however the converse is not true in general. In this paper, we will show that, if $E\neq \mathbb{P}^2$, then $A$ has a fat point module if and only if the quantum projective plane $\mathsf{Proj}_{\rm nc} A$ is finite over its center in the sense of this paper if and only if $|ν^*σ^3|<\infty$ where $ν$ is the Nakayama automorphism of $A$.In particular, we will show that if the second Hessian of $E$ is zero, then $A$ has no fat point module.

math.RA↗

arXiv2020

Hochschild cohomology related to graded down-up algebras with weights $(1,n)$

Let $A=A(α, β)$ be a graded down-up algebra with $({\rm deg}\,x, {\rm deg}\,y)=(1,n)$ and $β\neq 0$, and let $\nabla A$ be the Beilinson algebra of $A$. If $n=1$, then a description of the Hochschild cohomology group of $\nabla A$ is known. In this paper, we calculate the Hochschild cohomology group of $\nabla A$ for the case $n \geq 2$. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of $A$ is different depending on whether $\left(\begin{smallmatrix} 1&0 \end{smallmatrix}\right)\left(\begin{smallmatrix} α&1 \\ β&0 \end{smallmatrix}\right)^n\left(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right)$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $n\geq 3$ in the context of Grothendieck groups.

math.RT↗

arXiv2019

Some infinitely generated non projective modules over path algebras and their extensions under Martin's Axiom

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective (Theorem 1.10). In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{ω_1}$ such that, when $K$ is a countable field, $\operatorname{\sf MA}_{\aleph_1}$ (Martin's Axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${\rm Ext}^1_{KQ}(M_{ω_1},KQ)=0$ (Theorem 2.11).

math.RT↗

arXiv2019

Defining relations of 3-dimensional quadratic AS-regular algebras

Classification of AS-regular algebras is one of the main interests in non-commutative algebraic geometry. Recently, a complete list of superpotentials (defining relations) of all $3$-dimensional AS-regular algebras which are Calabi-Yau was given by Mori-Smith (the quadratic case) and Mori-Ueyama (the cubic case), however, no complete list of defining relations of all $3$-dimensional AS-regular algebras has not appeared in the literature. In this paper, we give all possible defining relations of $3$-dimensional quadratic AS-regular algebras. Moreover, we classify them up to isomorphism and up to graded Morita equivalence in terms of their defining relations in the case that their point schemes are not elliptic curves. In the case that their point schemes are elliptic curves, we give conditions for isomorphism and graded Morita equivalence in terms of geometric data.

math.RA↗

arXiv2014

On Hochschild cohomology of a self-injective special biserial algebra obtained by a circular quiver with double arrows

We calculate the dimensions of the Hochschild cohomology groups of a self-injective special biserial algebra $Λ_{s}$ obtained by a circular quiver with double arrows. Moreover, we give a presentation of the Hochschild cohomology ring modulo nilpotence of $Λ_{s}$ by generators and relations. This result shows that the Hochschild cohomology ring modulo nilpotence of $Λ_{s}$ is finitely generated as an algebra.

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