Growth of generalized Temperley–Lieb algebras connected with simple graphs

Abstract

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs ${\tilde A_n}$, ${\tilde D_n}$, ${\tilde E_6}$, or ${\tilde E_7}$. An algebra $T{L_{\Gamma, \tau }}$ has exponential growth if and only if the graph Γ coincides with none of the graphs ${A_n}$, ${D_n}$, ${E_n}$, ${\tilde A_n}$, ${\tilde D_n}$, ${\tilde E_6}$, and ${\tilde E_7}$