In this paper, we compute all possible differential structures of a $3$-dimensional DG Sklyanin algebra $mathcal{A}$, which is a connected cochain DG algebra whose underlying graded algebra $mathcal{A}^{#}$ is a $3$-dimensional Sklyanin algebra $S_{a
,b,c}$. We show that there are three major cases depending on the parameters $a,b,c$ of the underlying Sklyanin algebra $S_{a,b,c}$: (1) either $a^2 eq b^2$ or $c eq 0$, then $partial_{mathcal{A}}=0$; (2) $a=-b$ and $c=0$, then the $3$-dimensional DG Sklyanin algebra is actually a DG polynomial algebra; and (3) $a=b$ and $c=0$, then the DG Sklyanin algebra is uniquely determined by a $3times 3$ matrix $M$. It is worthy to point out that case (2) has been systematically studied in cite{MGYC} and case (3) is just the DG algebra $mathcal{A}_{mathcal{O}_{-1}(k^3)}(M)$ in cite{MWZ}. We solve the problem on how to judge whether a given $3$-dimensional DG Sklyanin algebra is Calabi-Yau.
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 theo
ry 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 $A$.
We study the representation theoretic results of the binary cubic generic Clifford algebra $mathcal C$, which is an Artin-Schelter regular algebra of global dimension five. In particular, we show that $mathcal C$ is a PI algebra of PI degree three an
d compute its point variety and discriminant ideals. As a consequence, we give a necessary and sufficient condition on a binary cubic form $f$ for the associated Clifford algebra $mathcal C_f$ to be an Azumaya algebra.
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.
The notion of $n$-th indicator for a finite-dimensional Hopf algebra was introduced by Kashina, Montgomery and Ng as gauge invariance of the monoidal category of its representations. The properties of these indicators were further investigated by Shi
mizu. In this short note, we show that the indicators appearing in positive characteristic all share the same sequence pattern if we assume the coradical of the Hopf algebra is a local Hopf subalgebra.
Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is
a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.
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.
For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive. In this pape
r, working over an algebraically closed field $mathbf{k}$ of prime characteristic $p$, we introduce the concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional connected Hopf algebras which are almost primitively generated; that is, these connected Hopf algebras are $p^{n+1}$-dimensional, whose primitive spaces are abelian restricted Lie algebras of dimension $n$. We illustrate this technique for the case $n=2$. Together with our preceding results in arXiv:1309.0286, we provide a complete classification of $p^3$-dimensional connected Hopf algebras over $mathbf{k}$ of characteristic $p>2$.
Let $p$ be a prime, $k$ be an algebraically closed field of characteristic $p$. In this paper, we provide the classification of connected Hopf algebras of dimension $p^3$, except the case when the primitive space of the Hopf algebra is two dimensiona
l and abelian. Each isomorphism class is presented by generators $x, y, z$ with relations and Hopf algebra structures. Let $mu$ be the multiplicative group of $(p^2+p-1)$-th roots of unity. When the primitive space is one-dimensional and $p$ is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by $A_{k}^1/mu$.
Let $p$ be a prime. We complete the classification on pointed Hopf algebras of dimension $p^2$ over an algebraically closed field $k$. When $text{char}k eq p$, our result is the same as the well-known result for $text{char}k=0$. When $text{char}k=p$
, we obtain 14 types of pointed Hopf algebras of dimension $p^2$, including a unique noncommutative and noncocommutative type.
