Universal central extensions of twisted forms of split simple Lie algebras over rings

We give sufficient conditions for the descent construction to be the universal central extension of a twisted form of a split simple Lie algebra over a ring. In particular, the universal central extensions of twisted multiloop Lie tori are obtained by the descent construction

Universal central extensions of slm∣nsl_{m|n} over Z/2ZZ/2Z-graded algebras

We study central extensions of the Lie superalgebra $sl_{m|n}(A)$, where $A$ is a $Z/2Z$-graded superalgebra over a commutative ring $K$. The Steinberg Lie superalgebra $st_{m|n}(A)$ plays a crucial role. We show that $st_{m|n}(A)$ is a central extension of $sl_{m|n}(A)$ for $m+n\geq 3$. We use a $Z/2Z$-graded version of cyclic homology to show that the center of the extension is isomorphic to $HC_1(A)$ as $K$-modules. For $m+n\geq 5$, we prove that $st_{m|n}(A)$ is the universal central extension of $sl_{m|n}(A)$. For $m+n=3,4$, we prove that $st_{2|1}(A)$ and $st_{3|1}(A)$ are both centrally closed. The universal central extension of $st_{2|2}(A)$ is constructed explicitly.Comment: 18 pages; section 7 added; reference [KT] added; the authors thank the referee for comments on the previous versio

Universal central extensions of direct limits of Lie superalgebras

We show that the universal central extension of a direct limit of perfect Lie superalgebras L_i is (isomorphic to) the direct limit of the universal central extensions of L_i. As an application we describe the universal central extensions of some infinite rank Lie superalgebras

Editorial Comment on the Special Issue of "Information in Dynamical Systems and Complex Systems"

This special issue collects contributions from the participants of the "Information in Dynamical Systems and Complex Systems" workshop, which cover a wide range of important problems and new approaches that lie in the intersection of information theory and dynamical systems. The contributions include theoretical characterization and understanding of the different types of information flow and causality in general stochastic processes, inference and identification of coupling structure and parameters of system dynamics, rigorous coarse-grain modeling of network dynamical systems, and exact statistical testing of fundamental information-theoretic quantities such as the mutual information. The collective efforts reported herein reflect a modern perspective of the intimate connection between dynamical systems and information flow, leading to the promise of better understanding and modeling of natural complex systems and better/optimal design of engineering systems