# Graph fusion algebras of ${\cal WLM}(p,p')$

#### by Jorgen Rasmussen

Institution: Mathematics and Statistics Dept., The University of Melbourne
Date: Mon 16th November 2009
Time: 1:00 PM
Location: Russell Love Theatre, Richard Berry Building, The University of Melbourne

Abstract: We consider the ${\cal W}$-extended logarithmic minimal
model ${\cal WLM}(p,p')$.
As in the rational minimal models, the so-called fundamental fusion algebra of ${\cal WLM}(p,p')$ is described by a simple graph fusion algebra. The fusion matrices in the regular representation thereof are mutually commuting, but in general not diagonalizable.
Nevertheless, we show that
they can be brought simultaneously to block-diagonal forms whose blocks are upper-triangular matrices of dimension 1, 3, 5 or 9. The graphs associated with the two fundamental modules are described in detail. The corresponding adjacency matrices share a complete set of common generalized eigenvectors organized as a web constructed by interlacing the Jordan chains of the two matrices. This web is here called a Jordan web and it consists of connected subwebs with 1, 3, 5 or 9 generalized eigenvectors. The similarity matrix, formed by concatenating these vectors, simultaneously brings the two fundamental adjacency matrices to Jordan canonical form modulo permutation similarity. The ranks of the participating Jordan blocks are 1 or 3, and the corresponding eigenvalues are given by $2\cos \frac{j\pi}{\rho}$ where $j=0,\ldots,\rho$ and $\rho=p,p'$.
For $p>1$, only some of the modules in the fundamental fusion algebra of ${\cal WLM}(p,p')$ are associated with boundary conditions within our lattice approach.
The fusion matrices in
the regular representation of the corresponding fusion subalgebra have features as above, but with the dimensions of the upper-triangular blocks and connected Jordan-web components given by 1, 2, 3 or 8.