具有负结构常数的融合代数
Fusion algebras with negative structure constants
摘要：
引入了结构常数不一定为正且不含单位元的融合代数。我们证明了它们在$\mathbb{C}$张量下是半单的，并且它们的特征满足正交关系.然后，我们定义了这类代数的子环和因子环的适当概念。对某些代数$R$，证明了存在一个具有非负结构常数的环$R'$，使得$R$是$R'$的因子环.给出了有限群的量子偶的表示环的有趣因子环的一些例子。然后，我们研究了与阿达玛矩阵相关的代数。对于一个$n\times n$ -矩阵，相应的代数是$\mathbb{Z}[{(\mathbb{Z}/2\mathbb{Z})}^{n-2}]$的一个子代数的因子环.
英文摘要：
We introduce fusion algebras with not necessarily positive structure constants and without identity element. We prove that they are semisimple when tensored with $\mathbb{C}$ and that their characters satisfy orthogonality relations. Then we define the proper notion of subrings and factor rings for such algebras. For certain algebras $R$ we prove the existence of a ring $R'$ with nonnegative structure constants such that $R$ is a factor ring of $R'$. We give some examples of interesting factor rings of the representation ring of the quantum double of a finite group. Then, we investigate the algebras associated to Hadamard matrices. For an $n\times n$-matrix the corresponding algebra is a factor ring of a subalgebra of $\mathbb{Z}[{(\mathbb{Z}/2\mathbb{Z})}^{n-2}]$.
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