2-Local automorphisms and derivations of triangular matrices

Let \(\mathcal {T}\) denote the algebra of all \(2 \times 2\) upper triangular matrices over a field \(\mathbb {F}\). We show that the linear space of all 2-local derivations on \(\mathcal {T}\) decomposes as \(\mathcal {L} = \mathcal {D} \oplus \mathcal {L}_0\), where \(\mathcal {D}\) is the subspace of all derivations, and \(\mathcal {L}_0\) consists of 2-local derivations vanishing on a subset of \(\mathcal {T}\), isomorphic to the space of functions \(f:\mathbb {F}\rightarrow \mathbb {F}\) such that \(f(0)=0\). For any 2-local automorphism \(\Lambda \) on \(\mathcal {T}\), we show that there exists a unique automorphism \(\phi \) and a 2-local automorphism \(\Lambda _{1} \in \varPsi \) such that \(\Lambda = \phi \Lambda _1\), where \(\varPsi \) is the monoid of 2-local automorphisms that act as the identity on a subset of \(\mathcal {T}\). Furthermore, we establish that \(\varPsi \) is isomorphic to the monoid of injective functions from \(\mathbb {F}^{*}\) to itself.