关于装饰关联代数的Peirce谱刚性
On the Peirce spectral rigidity of decorated incidence algebras
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中文总结 AI 辅助
本文研究由带符号装饰的部分Steiner三元系确定的代数，证明其幂等元谱仅依赖于负Pasch构型数，并给出显式矩阵模型展示刚性，联系极代数与对称Clifford系统。
中文摘要 AI 辅助
度量交换非结合代数中幂等元的乘法算子是一个自伴线性自同态，其谱反映了该代数所编码的代数与几何结构。尽管对于一般代数，幂等元的谱可以任意表现，但对于最有趣的类——其中包括几何中出现的Clifford型和Jordan型代数——谱受到高度约束。我们考虑由部分Steiner三元系（PSTS）确定的一族代数，其区组带有符号装饰，每个区组确定一个幂等元；即使对于最简单的PSTS，所得代数也可能相当复杂。对于Grassmannian PSTS $G_2(m)$的装饰，其区组是$m$元集合的$3$子集，其点参数化了一个底层的关联几何，谱表现为刚性：区组幂等元的带重数谱是单个组合参数——包含该区组的负号Pasch构型（四边形子构型）的数量——的函数。所得的区组谱组织成一个谱轮廓，通过二项式反演从矩不变量的层级中恢复。对于一类特殊的装饰，相关代数是满足独立于$m$的融合律的轴向代数，并且对于这些代数，给出了一个替代的显式矩阵模型直接展示刚性；该模型的偶部分是包含一个极代数直和项的直和，将该构造与极代数和对称Clifford系统联系起来。
英文摘要
The multiplication operator of an idempotent in a metrized commutative nonassociative algebra is a self-adjoint linear endomorphism whose spectrum reflects the algebraic and geometric structure encoded in the algebra. Although for general algebras the spectra of idempotents can behave arbitrarily, for the most interesting classes -- among them algebras of Clifford and Jordan type arising in geometry -- the spectrum is highly constrained. We consider a family of algebras determined by a partial Steiner triple system (PSTS) with blocks decorated by signs, each block determining an idempotent; even for the simplest PSTS the resulting algebras can be quite complicated. For decorations of the Grassmannian PSTS $G_2(m)$, whose blocks are $3$-subsets of an $m$-element set and whose points parametrize an underlying incidence geometry, the spectra behave rigidly: the spectrum with multiplicity of a block idempotent is a function of a single combinatorial parameter, the number of negatively signed Pasch configurations -- quadrilateral subconfigurations -- containing the block. The resulting block spectra organize into a spectral profile, recovered by binomial inversion from a hierarchy of moment invariants. For a distinguished family of decorations, the associated algebras are axial algebras satisfying a fusion law independent of $m$, and for these algebras there is given an alternative explicit matrix model exhibiting the rigidity directly; the even part of this model is a direct sum containing a polar algebra summand, linking the construction to polar algebras and symmetric Clifford systems.
发表机构
- Universidad Politécnica de Madrid（马德里理工大学）
- Linköping University（林雪平大学）
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