In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring Bˆ(G) of a finite group G is defined. An element of it is represented by a finite G-set with a G-equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic G-manifold with a G-equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of Bˆ(G). It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibers of Berglund–Hübsch dual invertible polynomials are enhanced dual to each other up to sign. As a byproduct, this implies the result about the orbifold zeta functions of Berglund–Hübsch–Henningson dual pairs obtained earlier.
We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: 1) this category has kernels and cokernels, 2) the cokernel is preserved under the passage to the group stereotype algebras, and 3) the notion of cokernel allows to prove that the continuous envelope Env C*(Z · K) of the group algebra of a compact buildup of an abelian locally compact group is an involutive Hopf algebra in the category of stereotype spaces (Ste, \odot). The last result plays an important role in the generalization of the Pontryagin duality for arbitrary Moore groups.
We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras.