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 $$ {\tilde E_7} $$ .