Natural transformations of the composition of Weil and cotangent functors (Q2759187)

For a Weil bundle \(T^A\) associated to a Weil algebra \(A\) and for a linear function \(f:A \to \mathbb R \), a natural transformation \(s_f :T^AT^* \to T^*T^A\) mapping the composition of a Weil bundle with the cotangent bundle to the comoposition of those natural bundles in the opposite order is defined. Further, the space \(S_A\) of them is described as a linear vector space isomorphic to the linear space \(A^*\) dual to \(A\). For Weil algebras of \((k,r)\)-velocities \(\mathbb D^r_k\) it is proved that the only cases for which there is a natural equivalence \(T^r_kT^* \to T^*T^r_k\) are exactly those for which \(k=1\). From this point of view, the natural equivalence \(T^r_1T^* \to T^*T^r_1\) by Cantrijn, Crampin, Sarlet and Saunders is described. For natural transformations \(s_f\) and natural vector fields \(X\) on \(T^A\), natural functions \(\varphi_{f, X}\) defined on \(T^AT^*\) are constructed and consequently any smooth combination of them. NEWLINENEWLINENEWLINEFor some cases of \(A\), namely \(\mathbb D^r_1\), \(\mathbb D^1_k\) and \(\mathbb D^1_1 \otimes \mathbb D^1_1\) it is proved that all natural functions defined on \(T^AT^*\) are given by this construction. Moreover, a procedure generating natural transformations \(T^AT^* \to T^*T^A\) is given. For basis elements \(f\) of \(A^*\), \(s_f\) play the role of basis natural transformations while their coefficients are natural functions defined on \(T^AT^*\). For some cases of \(A\), namely \(\mathbb D^1_k\), \(\mathbb D^2_1\) and \(\mathbb D^1_1 \otimes \mathbb D^1_1\), all natural transformations under disscussion are obtained by this procedure. Further, there are some relations between natural transformations \(s_f\) and liftings of \(1\)-forms and \((0,2)\)-tensor fields to Weil bundles.