Pseudo-topological commutative superalgebras with nilpotent ghosts (Q1813383)

Let \(\Lambda = \Lambda_ 0 \oplus \Lambda_ 1\) be a pseudotopological commutative superalgebra over \(\mathbf R\) with a Hausdorff topology, \(\Lambda_ 0 = {\mathbf R} \oplus G\), where \({\mathbf R} = {\mathbf R} \cdot e\) and \(G\) is a subalgebra (which is called the subalgebra of even souls). The author constructs a class of such algebras \(\Lambda\) in which all even souls are nilpotent and \(^ \perp \Lambda_ 1 = \{a \in \Lambda : a\Lambda_ 1 = 0\}\) is equal to zero. In some ways this class is useful for superanalysis on superspaces. Furthermore, generalized functions on superspaces over these commutative superalgebras are considered. It is shown that there exist fundamental solutions of a rather large class of linear differential operators with constant coefficients on superspaces. Note that this paper is a continuation of the author's research in functional superanalysis [see, e.g., Russ. Math. Surv. 43, No. 2, 103-137 (1988); translation from Usp. Mat. Nauk 43, No. 2(260), 87-114 (1988; Zbl 0665.46031)].