Cartan calculus for dual forms (Q2784566)

In [\textit{V. M. Buchstaber} (ed.) et al., Transl., Ser. 2, Am. Math. Soc. 179(33), 153-171 (1997; Zbl 0893.58008)] the author investigated forms on supermanifolds defined as Lagrangians of formal systems of equations (copaths), which may or may not define submanifolds, and defined an exterior differential in terms of variational derivatives with respect to a copath and established its main properties. NEWLINENEWLINENEWLINEIn this paper, the author considers a superspace \(V=V_0\oplus V_1\) of dimension \(n|m\) identified with a linear supermanifold. If \({\text{Vol}} V\) denotes the space of volume forms on \(V\), then a smooth map \({\mathcal L}\:V^*\times\dots\times V^*\times V^*\Pi\times\dots\times V^*\Pi\to{\text{Vol}} V\), with \(p\) \(V^*\) factors and \(q\) \(V^*\Pi\) factors, satisfying \({\mathcal L}(ph)={\mathcal L}(p){\text{Ber}} h\) for all \(h\in{\text{GL}}(p|q)\) and a certain condition on the second derivatives of \({\mathcal L}\), is called a dual form on \(V\) of bidegree \(p|q\). When \(V\) is replaced by \(W=V\oplus\mathbb R^{r|s}\), the author defines, in similar way, the notion of a mixed form on \(V\) and some operations on dual and mixed forms. The author states that these operations are stable, and then presents analogues of the Leibniz and homotopy identities for them.