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Interpreting superanalyticity in terms of convergent series. (English) Zbl 0676.58015

The notion of a superanalytic superfield (or a superfunction) is a key one within physical theories related to the grand unification. Superanalyticity can be defined in a way similar to the ordinary analyticity via the simple \((=pointwise)\) convergence for certain concrete supernumber algebras \((=\text{grad}ed\) commutative topological algebras of coefficients), such as Banach-Grassmann algebras [A. Jadczyk and K. Pilch, Commun. Math. Phys. 78, 373-390 (1981; Zbl 0464.58006)] and the DeWitt supernumber algebra [Sh. Matsumoto and K. Kakazu, J. Math. Phys. 27, 2690-2692 (1986; Zbl 0612.58005)].
An attempt to generalize this approach to more wide classes of ground algebras [A. Yu. Khrennikov, Russ. Math. Surv. 43, No.2, 103-137 (1988); translation from Usp. Mat. Nauk 43, No.2(260), 87-114 (1988; Zbl 0665.46031)] is totally unsatisfactory, as it is shown in an example concluding the reviewed note. Usually the superanalyticity is defined in terms of the so called superfield expansion [J. M. Rabin and L. Crane, Commun. Math. Phys. 100, 141-160 (1985; Zbl 0576.58004)], so that a particular choice of the coordinate system is involved. In the paper under review it is shown how one can avoid this.
A superanalytic superfunction is defined locally as a sum of a series formed by superpolynomials of a strictly growing order, convergent with respect to the newly introduced topology of supersimple convergence, which is in general strictly finer than that of a simple convergence. It is defined via the functor of points [cf. D. A. Leites, Theory of supermanifolds (1983; Zbl 0599.58001)].
Reviewer: V.Pestov

MSC:

58C50 Analysis on supermanifolds or graded manifolds
46G20 Infinite-dimensional holomorphy
30G30 Other generalizations of analytic functions (including abstract-valued functions)
16W50 Graded rings and modules (associative rings and algebras)
15A75 Exterior algebra, Grassmann algebras
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