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McKay matrices for finite-dimensional Hopf algebras

Published online by Cambridge University Press:  08 February 2021

Georgia Benkart*
Department of Mathematics, University of Wisconsin-Madison, Madison, WI53706, USA
Rekha Biswal
Max Planck Institute for Mathematics, 53111Bonn, Germany e-mail:
Ellen Kirkman
Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, NC27109, USA e-mail:
Van C. Nguyen
Department of Mathematics, United States Naval Academy, Annapolis, MD21402, USA e-mail:
Jieru Zhu
Department of Mathematics, University at Buffalo, Buffalo, NY14260-2900, USA e-mail:


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.

© Canadian Mathematical Society 2021

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