Some Non-Koszul Algebras

Abstract

The aim of this note is to show that the quadratic algebras ε_nstudied by S. Fomin and A. N. Kirillov in [FK] arenotKoszul algebras for anyn≥ 3. The algebraε _n (of type A) has generators T _ij for 1 ≤i≤j≤nsubject to the following relations:

$$\tau _{{ij}}^{2} = 0{\text{ for }}i < j; $$

(i)

$$\begin{array}{*{20}{c}} {{{\tau }_{{ij}}}{{\tau }_{{jk}}} = {{\tau }_{{jk}}}{{\tau }_{{ik}}} + {{\tau }_{{ik}}}{{\tau }_{{ij}}},} \hfill \\ {{{\tau }_{{jk}}}{{\tau }_{{ij}}} = {{\tau }_{{ik}}}{{\tau }_{{jk}}} + {{\tau }_{{ij}}}{{\tau }_{{ik}}} for i < j < k;} \hfill \\ \end{array} $$

(ii)

$${{\tau }_{{ij}}}{{\tau }_{{kl}}} = {{\tau }_{{kl}}}{{\tau }_{{ij}}}{\text{ whenever }}\{ i,j\} \cap \{ k,l\} {\text{ = }}\phi {\text{, }}i < j,{\text{ and }}k < l. $$

(iii)