本原轴代数中根相等性的一个二维反例
A Two-Dimensional Counterexample to Radical Equality in Primitive Axial Algebras
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中文总结 AI 辅助
针对本原轴代数中R(A,X)是否总等于J(A)的问题，本文构造特征非2域上的二维本原轴代数，证明二者不等，给出该问题的反例。
中文摘要 AI 辅助
设R(A,X)为本原轴代数(A,X)中不含指定生成集X中任何轴的最大理想，J(A)为A的极大理想之交。Mamontov、Shpectorov与Zhelyabina提出R(A,X)是否总等于J(A)的问题，我们给出否定答案。在所有特征非2的域上，基为a、b且乘法规则为a²=a、ab=2b、b²=b的二维交换代数，是满足显式融合律且生成集X={a,b}的本原轴代数。其完全理想格为0<Fb<A，故R(A,X)=0，J(A)=Fb，特别地，本原轴b属于J(A)。在复数域C上，该轴表示等价于此前已分类的D(-1),{e₂,a₆}表示，融合律为F_D3。因此该代数与轴结构为已知，新贡献在于计算其Jacobson根并得到根相等性问题的反例。
英文摘要
Let $R(A,X)$ denote the largest ideal of a primitive axial algebra $(A,X)$ that contains no axis from the specified generating set $X$, and let $J(A)$ be the intersection of the maximal ideals of $A$. Mamontov, Shpectorov, and Zhelyabin asked whether $R(A,X)=J(A)$ always holds. We give a negative answer. Over every field of characteristic different from $2$, the two-dimensional commutative algebra with basis $a,b$ and multiplication $a^2=a$, $ab=2b$, and $b^2=b$ is a primitive axial algebra for an explicit fusion law and the generating set $X={a,b}$. Its complete ideal lattice is $0<\mathbb{F}b<A$, whence $R(A,X)=0$ and $J(A)=\mathbb{F}b$; in particular, the primitive axis $b$ lies in $J(A)$. Over $\mathbb{C}$, this axial presentation is equivalent to the previously classified $D(-1),{e_2,a_6}$ presentation with fusion law $F_{D3}$. Thus the algebra and axial structure are known; the new point is the computation of its Jacobson radical and the resulting counterexample to the radical-equality question.
发表机构
- Institute of Mathematical Sciences, ShanghaiTech University（上海科技大学数学科学研究所）
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