McKay Matrices for Finite-dimensional Hopf Algebras

TL;DR

This paper studies the spectral properties of McKay matrices associated with finite-dimensional Hopf algebras, relating eigenvalues and eigenvectors to characters, and illustrates these results with the Drinfeld double of the Taft algebra using Chebyshev polynomials.

Contribution

It establishes new relationships between McKay matrices, projective McKay matrices, and characters for finite-dimensional Hopf algebras, with explicit Chebyshev polynomial expressions for the Drinfeld double of the Taft algebra.

Findings

Eigenvalues and eigenvectors are related to characters.

Explicit Chebyshev polynomial formulas for the Drinfeld double of the Taft algebra.

Spectral analysis of fusion rule matrices for Hopf algebras.

Abstract

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.