Generalized Burnside-Grothendieck ring functor and aperiodic ring functor associated with profinite groups

For every profinite group $G$, we construct two covariant functors $\Delta_G$ and ${\bf {\mathcal {AP}}}_G$ from the category of commutative rings with identity to itself, and show that indeed they are equivalent to the functor $W_G$ introduced in [A. Dress and C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, {\it Adv. in Math.} {\bf{70}} (1988), 87-132]. We call $\Delta_G$ the generalized Burnside-Grothendieck ring functor and ${\bf {\mathcal {AP}}}_G$ the aperiodic ring functor (associated with $G$). In case $G$ is abelian, we also construct another functor ${\bf Ap}_G$ from the category of commutative rings with identity to itself as a generalization of the functor ${\bf Ap}$ introduced in [K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, {\it Adv. in Math.} {\bf 81} (1990), 1-29]. Finally it is shown that there exist $q$-analogues of these functors (i.e, $W_G, \Delta_G, {\bf {\mathcal {AP}}}_G$, and ${\bf Ap}_G$) in case $G=\hat C$ the profinite completion of the multiplicative infinite cyclic group.Comment: minor corrections, 35 page

Necklace rings and logarithmic functions

In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor $Nr$ from the category of special \ld-rings into the category of special \ld-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper (J. Algebra. 253 (2002); no.1, 167-188) to the case of graded Lie (super)algebras with a group action by applying the Euler-Poincar\'e principle

q-Deformed necklace rings and q-Möbius function

AbstractWe introduce a q-deformation of the classical Möbius function and investigate its properties in connection with q-deformed truncated necklace rings. Also, we study the strictly natural isomorphism of q-deformed necklace rings