一维霍普夫代数的McKay矩阵
McKay Matrices for Finite-dimensional Hopf Algebras
摘要：
对于有限维Hopf代数$A$，一个$A$ -模$V$的McKay矩阵$M_V$编码了用$V$张紧简单$A$ -模的关系.我们证明了$M_V$的特征值和左、右（广义）特征向量的结果，将它们与字符联系起来。证明了用$V$张紧$A$的投射不可分解模得到的投射McKay矩阵$Q_V$与$V$的对偶模的McKay矩阵的关系.我们通过导出$M_V$和$Q_V$的特征值和特征向量的表达式，用几种切比雪夫多项式来说明Taft代数的Drinfeld双$D_n$的这些结果.对于矩阵$N_V$，编码的融合规则张$V$与投影不可分解的$D_n$ -模块的图像的Cartan映射的基础上，我们表明，特征值和特征向量也有这样的Chebyshev表达式。
英文摘要：
For a finite-dimensional Hopf algebra $A$, the McKay matrix $M_V$ of an $A$-module $V$ encodes the relations for tensoring the simple $A$-modules with $V$. We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $M_V$ by relating them to characters. We show how the projective McKay matrix $Q_V$ obtained by tensoring the projective indecomposable modules of $A$ with $V$ is related to the McKay matrix of the dual module of $V$. We illustrate these results for the Drinfeld double $D_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $M_V$ and $Q_V$ in terms of several kinds of Chebyshev polynomials. For the matrix $N_V$ that encodes the fusion rules for tensoring $V$ with a basis of projective indecomposable $D_n$-modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.
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