Leibniz algebras associated with representations of filiform Lie algebras

In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra $n_{n,1}$. We introduce a Fock module for the algebra $n_{n,1}$ and provide classification of Leibniz algebras $L$ whose corresponding Lie algebra $L/I$ is the algebra $n_{n,1}$ with condition that the ideal $I$ is a Fock $n_{n,1}$-module, where $I$ is the ideal generated by squares of elements from $L$.We also consider Leibniz algebras with corresponding Lie algebra $n_{n,1}$ and such that the action $I\times n_{n,1}\to I$ gives rise to a minimal faithful representation of $n_{n,1}$. The classification up to isomorphism of such Leibniz algebras is given for the case of $n=4$.     

Commutative n-ary superalgebras with an invariant skew-symmetric form

We study $n$-ary commutative superalgebras and $L_{\infty }$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative $m$-dimensional $(m-3)$-ary algebras with an invariant form, and a classification of real simple $m$-dimensional Lie $(m-3)$-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for $L_{\infty }$-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric $n$-ary algebras.     

Percolation on infinitely ramified fractal networks

We study how fractal features of an infinitely ramified network affect its percolation properties. The fractal attributes are characterized by the Hausdorff ($D_H$), topological Hausdorff ($D_{tH}$), and spectral ($d_s$) dimensions. Monte Carlo simulations of site percolation were performed on pre-fractal standard Sierpiński carpets with different fractal attributes. Our findings suggest that within the universality class of random percolation the values of critical percolation exponents are determined by the set of dimension numbers ($D_H$, $D_{tH}$, $d_s$), rather than solely by the spatial dimension (d). We also argue that the relevant dimension number for the percolation threshold is the topological Hausdorff dimension $D_{tH}$, whereas the hyperscaling relations between critical exponents are governed by the Hausdorff dimension $D_H$. The effect of the network connectivity on the site percolation threshold is revealed.     

A physical pathway to understand individual's labeling behavior in signed social networks

In this paper we propose a reshuffling approach to empirical analyze individual's labeling behavior in signed social networks. In our approach, each individual is assumed to have the ability to re-label his/her neighbors randomly with the parameters $p_s$ and $p_{+}$. Many reshuffled networks, which have the same topological structure and different signs' configuration, are built through applying our approach to the given three signed social networks. The entropy $S_{out}$ and the giant component ${\rho }_G$ for each reshuffled networks are calculated and analyzed. We find that there exist two kinds of individual's labeling behavior according to the suppressed effect of $S_{out}$ and the exponent α in the relationship of ${\rho }_G$ and $q_{+}$. Additionally, the suppressed effect of $S_{out}$ shows the non-randomness factor in individual's labeling behavior. These results offer new insights to understand human's behavior in online social networks.     

Infinitesimal supersymmetries over Lie groups and their classification for gl(1,1) and sl(1,1)

Infinitesimal supersymmetries over classical Lie groups that are not necessarily induced by a Lie supergroup are described. They yield a notion of supersymmetry that is less rigid than the assumption of a Lie supergroup action but still implies an underlying action of a Lie group. In contrast to Lie supergroups, the arising representation-theoretical Lie supergroups (RTLSG) occur as families associated to Harish–Chandra superpairs. However morphisms of RTLSGs directly correspond to morphisms of Harish–Chandra superpairs. Particular RTLSGs can be derived from the explicit constructions of Lie supergroups given by Kostant and Koszul. The Lie superalgebras $\mathfrak{g}\mathfrak{l}(1,1)$ or $\mathfrak{s}\mathfrak{l}(1,1)$ appearing also in higher dimensional classical Lie superalgebras, provide interesting first examples of RTLSGs. A classification of RTLSGs associated to real and complex $\mathfrak{g}\mathfrak{l}(1,1)$- and $\mathfrak{s}\mathfrak{l}(1,1)$-Harish–Chandra superpairs is given by parameter spaces and complete sets of invariants. The underlying Lie group is assumed to be connected but possibly not simply connected.