On minimal varieties of quadratic growth

In this paper we investigate the polynomial identities of an important subalgebra of the PI-algebra UT3 of 3 × 3 upper triangular matrices in characteristic zero. Moreover we prove that the five algebras which were used in Giambruno and La Mattina [A. Giambruno, D. La Mattina, PI-algebras with slow codimension growth, J. Algebra 284 (2005) 371–391] to classify (up to PI-equivalence) the algebras whose sequence of codimensions is bounded by a linear function generate the only five minimal varieties of quadratic growth.