Structures of Nichols (braided) Lie algebras of diagonal type

Let $V$ be a braided vector space of diagonal type. Let $\mathfrak B(V)$, $\mathfrak L^-(V)$ and $\mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $\mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $\mathfrak L(V)$ of arithmetic root systems and the dimension for $\mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $\mathfrak B(V) = F\oplus \mathfrak L^-(V)$ and $\mathfrak L^-(V)= \mathfrak L(V)$. We obtain an explicit basis of $\mathfrak L^ - (V)$ over quantum linear space $V$ with $\dim V=2$.Comment: 23 pages. Version to appear in Journal of Lie Theor

Assessing the habitat suitability of 10 serious weed species in global croplands

Abstract Weed expansion has the potential to severely affect global agriculture. Habitat suitability models (HSMs) have been widely used to assess weed expansion to enable effective prevention and control of weeds. However, few studies have focused on the habitat suitability of global croplands for these weeds. Previous studies listed the 10 serious weed species that have the largest negative impact on global agriculture. Here, we used Maxent modeling (an HSM with good performance) to quantify the habitat suitability of global croplands for these 10 weed species using occurrence records and environmental variables (i.e., climate factors, soil properties, and human footprint). We compared the habitat suitability values of these 10 serious weed species in 174 different types of cropland. The habitat suitability values for all the weed species studied was high in global croplands. Furthermore, habitat suitability may depend on the type of cropland and the spatial variation created by varying climate factors, soil properties, and human footprint. Cynodon dactylon, Echinochloa crus-galli, Eleusine indica, Panicum maximum, and Sorghum halepense had the highest habitat suitability values in chicory plantations. Cyperus rotundus and Echinochloa colona had the highest habitat suitability values in olive plantations. The habitat suitability values for Imperata cylindrica was the highest in mustard plantations, Eichhornia crassipes was most suited to expansion in raspberry plantations, and the habitat suitability values for Lantana camara was the highest in chickpea plantations. Risk prevention and control should be based on the cropland type for these 10 serious weed species, taking into account climate factors, soil properties, and human activities. Our study provides guidelines for effective management of weed risk in different croplands globally