Let L be a Lie algebra acting by derivations on an associative algebra A over a field F of characteristic zero. The polynomial identities satisfied by A with respect to this action are called differential identities, or L-identities. In this paper, we study the differential identities of the algebra \( UT_k \) of k×k upper triangular matrices and take a first step toward the classification of minimal L-varieties of differential exponent 3.
We first prove that, whenever \( UT_k \) generates a minimal variety of algebras with derivations, the L-action can be replaced by its semisimple part. More precisely, it is enough to consider inner derivations induced by diagonal elements. We then apply this reduction to \( UT_3 \) and explicitly determine the TL-ideal of differential identities and the corresponding differential codimension sequence for every such action on \( UT_3 \). Finally, we show that every L-variety generated by \( UT_k \), with k≥3, contains \( UT_3 \) endowed with one of these L-actions.