Necklace rings and logarithmic functions

On the structure of the necklace Lie algebra

In this note, we initiate the systematic study of the Lie algebra structure of the necklace Lie algebra n of a free algebra in 2d variables. We begin by giving a description of n as an sp(2d)-module. Specializing to d = 1, we decompose n into a direct sum of highest weight modules for sl_2, the coefficients of which are given by a closed formula. Next, we observe that n has a nontrivial center, which we link through the center C of the trace ring of couples of generic 2x2 matrices to the Poisson center of S(sl_2). The Lie algebra structure of n induces a Poisson structure on C, the symplectic …

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

For every profinite group $G$, we construct two covariant functors $Δ_G$ 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 $Δ_G$ 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 ${\…

赫维茨级数环中的形变卷积、累积量变换与半群生成元

Deformed Convolution, Cumulant Transforms, and Semigroup Generators in Hurwitz Series Rings

该研究在赫维茨级数环上引入单参数卷积，推导其性质与应用，得到相关半群生成元的显式公式，拓展了自由概率与代数结构的相关理论。