首页/论文/1908.01430

arXiv 1908.014302019-08math.RA

A note on generic Clifford algebras of binary cubic forms

Linhong Wang, Xingting Wang

备注：: to appear Algebr. Represent. Theory

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Classifying Quadratic Quantum P^2s by using Graded Skew Clifford Algebras

We prove that quadratic regular algebras of global dimension three on degree-one generators are related to graded skew Clifford algebras. In particular, we prove that almost all such algebras may be constructed as a twist of either a regular graded skew Clifford algebra or of an Ore extension of a regular graded skew Clifford algebra of global dimension two. In so doing, we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in P^2.

2011-06-08math.RA

On representations of Clifford algebras of ternary cubic forms

In this article, we provide an overview of a one-to-one correspondence between representations of the generalized Clifford algebra $C_f$ of a ternary cubic form $f$ and certain vector bundles (called Ulrich bundles) on a cubic surface $X$. We study general properties of Ulrich bundles, and using a recent classification of Casanellas and Hartshorne, deduce the existence of irreducible representations of $C_f$ of every possible dimension.

2011-07-07math.AGmath.RA

Minimal cubic cones via Clifford algebras

We construct two infinite families of algebraic minimal cones in $R^{n}$. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any $n\ge4$, $n\ne 16k+1$, there is at least one minimal cone in $R^{n}$ given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in $R^{m^2}$, $m\ge2$, defined by an irreducible homogeneous polynomial of degree $…

2010-10-10math.CVmath.DGmath.RA

A note on solution of $ax+xb=c$ by Clifford algebras

The coordinate-free solutions of the multivector equation $ax+xb=c$ are discussed and presented for the Clifford algebras $Cl_{p,q}$ when $p+q\le 3$.

2019-02-25math-phmath.MPmath.RA

关于 Clifford 代数中的中心化子与扭曲中心化子的注记

A Note on Centralizers and Twisted Centralizers in Clifford Algebras

本文研究了退化与非退化 Clifford 代数中的中心化子与扭曲中心化子，给出了固定次数子空间及其直和的显式形式，结果可应用于计算机科学、物理学和工程领域。

2024-08-24math-phmath.MPmath.RA

关于Clifford丛与某些Finsler型空间的注记

A note on Clifford bundles and certain Finsler type spaces

本文研究Clifford丛扩张与Finsler型结构的关系，证明在平坦度量下，由（伪）黎曼度量和1-形式构造的Finsler结构、Clifford丛扩张及Dirac型算子之间存在三角映射。

2021-06-01hep-thmath-phmath.DG

更多相关论文收起更多论文

Abelian Group Clifford Algebras

In "A note on generalized Clifford algebras and representations" (Caenepeel, S.; Van Oystaeyen, F., Comm. Algebra 17 (1989) no. 1, 93--102.) generalized Clifford algebras were introduced via Clifford representations; these correspond to projective representations of a finite group (Abelian), $G$ say, such that the corresponding twisted group ring has minimal center. The latter then translates to the fact that the corresponding 2-cocycle allows a minimal (none!) number of ray classes and this forces a decomposition of $G$ in cyclic components in a suitable way, cf. Zmud, M., Symplectic geometri…

2009-03-26math.RA

Graded Morita equivalence of Clifford superalgebras

This note uses a variation of graded Morita theory for finite dimensional superalgebras to determine explicitly the graded basic superalgebras for all real and complex Clifford superalgebras. As an application, the Grothendieck groups of the category of left $\mathbb{Z}_2$-graded modules over all real and complex Clifford superalgebras are described explicitly.

2012-04-19math.RAmath.RT

A moduli interpretation of untwisted binary cubic forms

We give a moduli interpretation to the quotient of (nondegenerate) binary cubic forms with respect to the natural $\text{GL}_2$-action on the variables. In particular, we show that these $\text{GL}_2$ orbits are in bijection with pairs of $j$-invariant $0$ elliptic curves together with $3$-torsion Brauer classes that are invariant under complex multiplication. The binary cubic generic Clifford algebra plays a key role in the construction of this correspondence.

2021-03-30math.AGmath.RA

关于斜二次超曲面偶Clifford代数的注记

A note on even Clifford algebras of skew quadric hypersurfaces

Shinshu University（信州大学）

本文研究斜二次超曲面的偶Clifford代数结构，证明其与矩阵代数等价，并展示其在Cohen-Macaulay模范畴中的三角等价性，揭示其作为非交换代数的自然推广。

2026-04-08math.RAmath.RT

Clifford algebra, geometric algebra, and applications

These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors…

2009-07-30math-phmath.MPmath.RA

关于交换的克莱因代数及其分解

On Commutative Analogues of Clifford Algebras and Their Decompositions

本文研究了交换的克莱因代数及其分解，通过共轭运算和幂等元等工具，提供了张量积和直接和分解方法。

2025-12-20math.ACmath.RA

Bott type periodicity for the higher octonions

We study the series of complex nonassociative algebras On and real nonassociative algebras $O_{p,q}$ introduced in [10]. These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras $O_{n}$ and $O_{p,q}$ with $p + q = n$ have a natural $Z_n^2$-grading, and they are characterized by cubic forms over the field $Z_2$. We establish a periodicity for the algebras $O_{n}$ and $O_{p,q}$ similar to that of the Clifford algebras Cln and $Cl_{p,q}$.

2014-05-26math.ACmath.RA

Introduction to Noncommutative Algebraic Geometry (First Draft)

This Lecture Notes is meant to introduce noncommutative algebraic geometry tools (which were invented by M. Artin, W. Schelter, J. Tate, and M. Van den Bergh in the late 1980s) and also graded skew Clifford algebras (which were introduced by T. Cassidy and M. Vancliff).

2015-02-18math.RA

Okubo代数与例外周期性：四元码分次、根格阻碍与Clifford锥

Okubo algebras and Exceptional Periodicity: tetracode gradings, root-lattice obstructions and Clifford cones

University of Hertfordshire（赫特福德大学）；Instituto Superior Técnico（里斯本高等技术学院）；Universidade do Algarve（阿尔加维大学）

该研究探讨Okubo代数的结构扩展至例外周期性代数，分析其分次、根格阻碍等性质，确定相关代数的自同构群，给出EP tip的Clifford模实现及超引力相关的齐次标量流形。

2026-10-14hep-thmath.RAmath.RT

Function Theories in Cayley-Dickson algebras and Number Theory

In the recent years a lot of effort has been made to extend the theory of hyperholomorphic functions from the setting of associative Clifford algebras to non-associative Cayley-Dickson algebras, starting with the octonions. An important question is whether there appear really essentially different features in the treatment with Cayley-Dickson algebras that cannot be handled in the Clifford analysis setting. Here we give one concrete example. Cayley-Dickson algebras namely admit the construction of direct analogues of CM-lattices, in particular lattices that are closed under multiplication. Can…

2019-12-03math.CVmath.NTmath.RA