McKay matrices for finite-dimensional Hopf algebras

Benkart, Georgia ; Biswal, Rekha ; Kirkman, Ellen ; Nguyen, Van C. ; Zhu, Jieru

Canadian journal of mathematics, Tome 74 (2022) no. 3, pp. 686-731

Résumé

For a finite-dimensional Hopf algebra $\mathsf {A}$, the McKay matrix $\mathsf {M}_{\mathsf {V}}$ of an $\mathsf {A}$-module $\mathsf {V}$ encodes the relations for tensoring the simple $\mathsf {A}$-modules with $\mathsf {V}$. We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $\mathsf {M}_{\mathsf {V}}$ by relating them to characters. We show how the projective McKay matrix $\mathsf {Q}_{\mathsf {V}}$ obtained by tensoring the projective indecomposable modules of $\mathsf {A}$ with $\mathsf {V}$ is related to the McKay matrix of the dual module of $\mathsf {V}$. We illustrate these results for the Drinfeld double $\mathsf {D}_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $\mathsf {M}_{\mathsf {V}}$ and $\mathsf {Q}_{\mathsf {V}}$ in terms of several kinds of Chebyshev polynomials. For the matrix $\mathsf {N}_{\mathsf {V}}$ that encodes the fusion rules for tensoring $\mathsf {V}$ with a basis of projective indecomposable $\mathsf {D}_n$-modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

Détail

DOI : 10.4153/S0008414X21000067

Mots-clés : McKay matrix, Drinfeld double, character, Chebyshev polynomial