On enlargability of infinite-dimensional Lie superalgebras (Q1801599)

One of the important goals of supermanifold theory is to provide means for ``enlarging'' a Lie superalgebra, by attributing to it a supergroup - - that is, group object in a category of supermanifolds. It is shown in the present work that in infinite dimensions a Lie superalgebra \(\mathfrak g\) coming from a Rogers supergroup may not come from a DeWitt one. Thus, we produce an evidence that a whole class of Lie superalgebras can be enlarged only by means of the most general ``naïve'' approach to supermanifolds, which therefore cannot be merely thrown away. Our conclusion is that the problem of rewording the ``naïve'' approach in the language of Berezin-Leĭtes-Kostant supermanifolds (possibly infinite-dimensional) and their deformations becomes substantial.