论文标题
McKay矩阵用于有限维的HOPF代数
McKay Matrices for Finite-dimensional Hopf Algebras
论文作者
论文摘要
对于有限维的Hopf代数$ a $，McKay Matrix $ m_v $ a $ a $ -module $ v $编码与$ v $的简单$ a $ modules的关系。我们通过将$ m_v $的特征值以及左右（广义）特征向量证明了$ m_v $的结果。我们展示了通过张贴$ a $ $ a $的投影不可分解的模块获得的投射mckay矩阵$ q_v $与$ v $的不可分解的模块与$ v $的双模块的mckay矩阵有关。我们通过在几种chebyshev polynomials中为$ m_v $的特征值和特征值和特征向量的表达方式来说明TAFT代数的Drinfeld Double $ d_n $。对于矩阵$ n_v $编码张贴$ v $的融合规则，其基础是cartan地图图像的投影不可分解的$ d_n $ modules，我们表明特征值和特征值也具有这样的chebyshev表达式。
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.
