Nijenhuis operators on Leibniz algebras

In this paper, we study Nijenhuis operators on Leibniz algebras. We discuss the relationship of Nijenhuis operators with Rota-Baxter operators and modified Rota-Baxter operators on Leibniz algebras. We define a representation theory of Nijenhuis Leibniz algebras and construct a cohomology theory. Next, we define a one-parameter formal deformation theory of Nijenhuis Leibniz algebras and study infinitesimals, rigidity, and equivalences along the line of Gerstenhaber deformation theory. As an application of our cohomology theory, we show that our cohomology is deformation cohomology and study abelian extensions of such algebras.Comment: 21 pages. arXiv admin note: text overlap with arXiv:2208.0056

Cohomology of modified Rota-Baxter Leibniz algebra of weight λ\lambda

Rota-Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota-Baxter operators on Leibniz algebras. We investigate modified Rota-Baxter Leibniz algebras from the cohomological point of view. We study a one-parameter formal deformation theory of modified Rota-Baxter Leibniz algebras and define the associated deformation cohomology that controls the deformation. Finally, as an application, we characterize equivalence classes of abelian extensions in terms of second cohomology groups.Comment: 20 page

Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras

A Rota-Baxter Leibniz algebra is a Leibniz algebra $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and formal deformation theory of Rota-Baxter Leibniz algebras and show that our cohomology is deformation cohomology. Moreover, We define an abelian extension of Rota-Baxter Leibniz algebras and show that equivalence classes of such extensions are related to the cohomology groups.Comment: 25 Page

On Nijenhuis Lie triple systems

In this paper, we investigate the mathematical structure of Nijenhuis Lie triple systems, an extension of classical Lie triple systems augmented with the Nijenhuis operator. Our study focuses on the cohomology of Nijenhuis Lie triple systems and demonstrates how abelian extensions of Nijenhuis Lie triple systems are related to cohomology groups. Additionally, we define Nijenhuis Lie triple 2-systems and also classify `strict' and `skeletal' Nijenhuis Lie triple 2-systems in terms of crossed modules and the cohomology of Nijenhuis Lie triple systems.Comment: 20 page

Arsenic mobilization in the aquifers of three physiographic settings of West Bengal, India: understanding geogenic and anthropogenic influences

A comparative hydrogeochemical study was carried out in West Bengal, India covering three physiographic regions, Debagram and Chakdaha located in the Bhagirathi-Hooghly alluvial plain and Baruipur in the delta front, to demonstrate the control of geogenic and anthropogenic influences on groundwater arsenic (As) mobilization. Groundwater samples (n = 90) from tube wells were analyzed for different physico-chemical parameters. The low redox potential (Eh = -185 to -86 mV) and dominant As(III) and Fe(II) concentrations are indicative of anoxic nature of the aquifer. The shallow (100 m) aquifers of Bhagirathi-Hooghly alluvial plains as well as shallow aquifers of delta front are characterized by Ca 2+ {single bond}HCO 3 - type water, whereas Na + and Cl - enrichment is found in the deeper aquifer of delta front. The equilibrium of groundwater with respect to carbonate minerals and their precipitation/dissolution seems to be controlling the overall groundwater chemistry. The low SO 4 2- and high DOC, PO 4 3- and HCO 3 concentrations in groundwater signify ongoing microbial mediated redox processes favoring As mobilization in the aquifer. The As release is influenced by both geogenic (i.e. geomorphology) and anthropogenic (i.e. unsewered sanitation) processes. Multiple geochemical processes, e.g., Fe-oxyhydroxides reduction and carbonate dissolution, are responsible for high As occurrence in groundwaters