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Xingting Wang

Publications and source records attributed to Xingting Wang.

At least 37 records · Page 2Linked to original sources
arXiv2021

Poisson geometry and representations of PI 4-dimensional Sklyanin algebras

Take S to be a 4-dimensional Sklyanin (elliptic) algebra that is module-finite over its center Z; thus, S is PI. Our first result is the construction of a Poisson Z-order structure on S such that the induced Poisson bracket on Z is non-vanishing. We also provide the explicit Jacobian structure of this bracket, leading to a description of the symplectic core decomposition of the maximal spectrum Y of Z. We then classify the irreducible representations of S by combining (1) the geometry of the Poisson order structures, with (2) algebro-geometric methods for the elliptic curve attached to S, along with (3) representation-theoretic methods using line and fat point modules of S. Along the way, we improve results of Smith and Tate obtaining a description the singular locus of Y for such S. The classification results for irreducible representations are in turn used to determine the zero sets of the discriminants ideals of these algebras S.

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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 $A$.

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arXiv2021

Reflection Groups and Rigidity of Quadratic Poisson Algebras

In this paper, we study the invariant theory of quadratic Poisson algebras. Let G be a finite group of the graded Poisson automorphisms of a quadratic Poisson algebra A. When the Poisson bracket of A is skew-symmetric, a Poisson version of the Shephard-Todd-Chevalley theorem is proved stating that the fixed Poisson subring A^G is skew-symmetric if and only if G is generated by reflections. For many other well-known families of quadratic Poisson algebras, we show that G contains limited or even no reflections. This kind of Poisson rigidity result ensures that the corresponding fixed Poisson subring A^G is not isomorphic to A as Poisson algebras unless G is trivial.

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arXiv2021

DG Algebra structures on the quantum affine $n$-space $\mathcal{O}_{-1}(k^n)$

Let $\mathcal{A}$ be a connected cochain DG algebra, whose underlying graded algebra $\mathcal{A}^{\#}$ is the quantum affine $n$-space $\mathcal{O}_{-1}(k^n)$. We compute all possible differential structures of $\mathcal{A}$ and show that there exists a one-to-one correspondence between $$\{\text{cochain DG algebra}\,\,\mathcal{A}\,|\,\mathcal{A}^{\#}=\mathcal{O}_{-1}(k^n)\}$$ and the $n\times n$ matrices $M_n(k)$. For any $M\in M_n(k)$, we write $\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ for the DG algebra corresponding to it. We also study the isomorphism problems of these non-commutative DG algebras. For the cases $n\le 3$, we check their homological properties. Unlike the case of $n=2$, we discover that not all of them are Calabi-Yau when $n=3$. In spite of this, we recognize those Calabi-Yau ones case by case. In brief, we solve the problem on how to judge whether a given such DG algebra $\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ is Calabi-Yau.

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arXiv2020

Poisson Dixmier-Moeglin equivalence from a topological point of view

In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for $A$ in terms of the poset $({\rm P. spec A}, \subseteq)$ and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O.L. Sánchez, and B. Moosa, Poisson algebras via model theory and differential algebraic geometry, J. Eur. Math. Soc. (JEMS), 19(2017), no. 7, 2019-2049] to the general context of a commutative differential algebra.

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arXiv2020

The Zariski cancellation problem for Poisson algebras

We study the Zariski cancellation problem for Poisson algebras asking whether $A[t]\cong B[t]$ implies $A\cong B$ when $A$ and $B$ are Poisson algebras. We resolve this affirmatively in the cases when $A$ and $B$ are both connected graded Poisson algebras finitely generated in degree one without degree one Poisson central elements and when $A$ is a Poisson integral domain of Krull dimension two with nontrivial Poisson bracket. We further introduce Poisson analogues of the Makar-Limanov invariant and the discriminant to deal with the Zariski cancellation problem for other families of Poisson algebras.

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arXiv2019

A note on generic Clifford algebras of binary cubic forms

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 and 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.

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arXiv2019

Nonsplit module extensions over the one-sided inverse of k[x]

Let $R$ be the associative $k$-algebra generated by two elements $x$ and $y$ with defining relation $yx=1$. A complete description of simple modules over $R$ is obtained by using the results of Irving and Gerritzen. We examine the short exact sequence $0\rightarrow U\rightarrow E \rightarrow V\rightarrow 0$, where $U$ and $V$ are simple $R$-modules. It shows that nonsplit extension only occurs when both $U$ and $V$ are one-dimensional, or, under certain condition, $U$ is infinite-dimensional and $V$ is one-dimensional.

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arXiv2019

FS-indicators of pq-dimensional pointed Hopf algebras

We compute higher Frobenius-Schur indicators of pq-dimensional pointed Hopf algebras in characteristic p through their associated graded Hopf algebras. These indicators are gauge invariants for the monoidal categories of representations of these algebras.

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arXiv2019

Gelfand-Kirillov dimension of cosemisimple Hopf algebras

In this note, we compute the Gelfand-Kirillov dimension of cosemisimple Hopf algebras that arise as deformations of a linearly reductive algebraic group. Our work lies in a purely algebraic setting and generalizes results of Goodearl-Zhang (2007), of Banica-Vergnioux (2009), and of D'Andrea-Pinzari-Rossi (2017).

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arXiv2019

A note on Masuoka's Theorem for semisimple irreducible Hopf algebras

Masuoka proved (2009) that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple if and only if it is commutative semisimple if and only if the Hopf subalgebra generated by all primitives is semisimple. In this paper, we give another proof of this result by using Hochschild cohomology of coalgebras.

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arXiv2019

Indicators of Hopf algebras in positive characteristic

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 Shimizu. 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.

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arXiv2019

Hopf algebras of prime dimension in positive characteristic

We prove that a Hopf algebra of prime dimension $p$ over an algebraically closed field, whose characteristic is equal to $p$, is either a group algebra or a restricted universal enveloping algebra. Moreover, we show that any Hopf algebra of prime dimension $p$ over a field of characteristic $q>0$ is commutative and cocommutative when $q=2$ or $p<4q$. This problem remains open in positive characteristic when $2<q<p/4$.

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arXiv2018

Poisson geometry of PI 3-dimensional Sklyanin algebras

We give the 3-dimensional Sklyanin algebras $S$ that are module-finite over their center $Z$ the structure of a Poisson $Z$-order (in the sense of Brown-Gordon). We show that the induced Poisson bracket on $Z$ is non-vanishing and is induced by an explicit potential. The ${\mathbb Z}_3 \times \Bbbk^\times$-orbits of symplectic cores of the Poisson structure are determined (where the group acts on $S$ by algebra automorphisms). In turn, this is used to analyze the finite-dimensional quotients of $S$ by central annihilators: there are 3 distinct isomorphism classes of such quotients in the case $(n,3) \neq 1$ and 2 in the case $(n,3)=1$, where $n$ is the order of the elliptic curve automorphism associated to $S$. The Azumaya locus of $S$ is determined, extending results of Walton for the case $(n,3)=1$.

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arXiv2018

The Dixmier-Moeglin equivalence, Morita equivalence, and homeomorphism of spectra

Let $k$ be a field and let $R$ be a left noetherian $k$-algebra. The algebra $R$ satisfies the Dixmier-Moeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the ${\rm Spec}(R)$ and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. We show that if $R$ and $S$ are two left noetherian $k$-algebras with ${\rm dim}_k(R), {\rm dim}_k(S)<|k|$ then if $R$ and $S$ have homeomorphic spectra then $R$ satisfies the Dixmier-Moeglin equivalence if and only if $S$ does. In particular, the topology of ${\rm Spec}(R)$ can detect the Dixmier-Moeglin equivalence in this case. In addition, we show that if $k$ is uncountable and $R$ is affine noetherian and its prime spectrum is a disjoint union of subspaces that are each homeomorphic to the spectrum of an affine commutative ring then $R$ satisfies the Dixmier-Moeglin equivalence. We show that neither of these results need hold if $k$ is countable and $R$ is infinite-dimensional. Finally, we make the remark that satisfying the Dixmier-Moeglin equivalence is a Morita invariant and finally we show that $R$ and $S$ are left noetherian $k$-algebras that satisfy the Dixmier-Moeglin equivalence then $R\otimes_k S$ does too, provided it is left noetherian and satisfies the Nullstellensatz; and we show that $eRe$ also satisfies the Dixmier-Moeglin equivalence, where $e$ is a nonzero idempotent of $R$.

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arXiv2017

The cohomology ring of some Hopf algebras

Let p be a prime, and k be a field of characteristic p. We investigate the algebra structure and the structure of the cohomology ring for the connected Hopf algebras of dimension p^3, which appear in the classification obtained in [V.C. Nguyen, L.-H. Wang and X.-T. Wang, Classification of connected Hopf algebras of dimension p^3, J. Algebra 424 (2015), 473-505]. The list consists of 23 algebras together with two infinite families. We identify the Morita type of the algebra, and in almost all cases this is sufficient to clarify the structure of the cohomology ring.

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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.

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