Identities with derivation of the algebra of upper triangular matrices of size two

Let UT2(F) be the algebra of 2  2 upper triangular matrices over a eld F of characteristic zero. We rst determine the Lie algebra of derivation of UT2(F) in order to study its dierential polynomial identities. We give a complete description of the space of multilinear dierential identities: on one hand we determine the generators of the corresponding T-ideal of the free algebra with derivation, while on the other we completely describe identities with derivation in the language of Young diagrams through the representation theory of the symmetric group Sn.