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We study structural, magnetic and magneto-thermal properties of GdRhIn compound. The room temperature X-ray diffraction measurements show hexagonal crystal structure. Temperature and field dependence of magnetization suggest two magnetic transitions antiferromagnetic to ferromagnetic at 16 K and ferromagnetic to paramagnetic at 34. The heat capacity measurements confirm both the magnetic transitions in GdRhIn. The magnetization data was used to calculate isothermal magnetic entropy change and refrigerant capacity in GdRhIn, which was found to be 10.3 J/Kg-K for the field change of 70 kOe and 282 J/Kg for the field change of 50 kOe, respectively. The large magnetocaloric effect in GdRhIn suggests that the material could be used for magnetic refrigeration at low temperatures., Comment: 11 pages, 4 figures

The notion of Poisson dialgebras was introduced by Loday. In this article, we propose a new definition with some modifications that is supported by several canonical examples coming from Poisson algebra modules, averaging operators on Poisson algebras, and differential Poisson algebras. We show that a Poisson object in the category of linear maps has an associated Poisson dialgebra structure. Conversely, starting from a Poisson dialgebra we describe a Poisson object in the category of linear maps. These constructions yield a pair of adjoint functors between the category of Poisson objects in the category of linear maps and the category of Poisson dialgebras. There is a Lie $2$-algebra associated with any Leibniz algebra. Here, we first obtain an associative $2$-algebra starting from a dialgebra. Then, for a Poisson dialgebra, we construct a graded space that inherits both a Lie $2$-algebra and an associative $2$-algebra structure. In a particular case of Poisson dialgebras, which we call `reduced Poisson dialgebra', we obtain an associated $2$-term homotopy Poisson algebra (of degree $0$)., Comment: Any comments and remarks are welcome

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of extensions of Lie-Yamaguti algebras. More precisely, given an abelian extension $$0 \to V \xrightarrow[]{i} \widetilde{L} \xrightarrow[]{p} L \to 0$$ of a Lie-Yamaguti algebra $L$, we are interested in finding the pairs $(\phi, \psi)\in \mathrm{Aut}(V)\times \mathrm{Aut}(L)$, which are inducible by an automorphism in $\mathrm{Aut}(\widetilde{L})$. We connect the inducibility problem to the $(2,3)$-cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphism in $\mathrm{Aut}(V)\times \mathrm{Aut}(L)$ to be inducible lies in the $(2,3)$-cohomology group $\mathrm{H}^{(2,3)}(L,V)$. We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and $(2,3)$-cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In the sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index $2$. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index $2$., Comment: Any comments/suggestions are welcome

A relative Rota-Baxter algebra is a generalization of a Rota-Baxter algebra. Relative Rota-Baxter algebras are closely related to dendriform algebras. In this paper, we introduce bimodules over a relative Rota-Baxter algebra that fits with the representations of dendriform algebras. We define the cohomology of a relative Rota-Baxter algebra with coefficients in a bimodule and then study abelian extensions of relative Rota-Baxter algebras in terms of the second cohomology group. Finally, we consider homotopy relative Rota-Baxter algebras and classify skeletal homotopy relative Rota-Baxter algebras in terms of the above-defined cohomology., Comment: Comments and suggestions are welcome

A Rota-Baxter Lie algebra $\mathfrak{g}_T$ is a Lie algebra $\mathfrak{g}$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. In this paper, we consider non-abelian extensions of a Rota-Baxter Lie algebra $\mathfrak{g}_T$ by another Rota-Baxter Lie algebra $\mathfrak{h}_S.$ We define the non-abelian cohomology $H^2_{nab} (\mathfrak{g}_T, \mathfrak{h}_S)$ which classifies {equivalence classes of} such extensions. Given a non-abelian extension $$ 0 \rightarrow \mathfrak{h}_S \xrightarrow{i} \mathfrak{e}_U \xrightarrow{p} \mathfrak{g}_T \rightarrow 0$$ of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of Rota-Baxter automorphisms in $\mathrm{Aut}(\mathfrak{h}_S ) \times \mathrm{Aut}(\mathfrak{g}_T)$ to be induced by an automorphism in $\mathrm{Aut}(\mathfrak{e}_U)$ lies in the cohomology group $H^2_{{nab}} (\mathfrak{g}_T, \mathfrak{h}_S)$. As a byproduct, we obtain the Wells short-exact sequence in the context of Rota-Baxter Lie algebras., Comment: Any comments/suggestions are welcome

Let $B$ be a commutative algebra and $A$ be a $B$-algebra (determined by an algebra homomorphism $\varepsilon:B\rightarrow A$). M. D. Staic introduced a Hochschild like cohomology $H^{\bullet}((A,B,\varepsilon);A)$ called secondary Hochschild cohomology, to describe the non-trivial $B$-algebra deformations of $A$. J. Laubacher et al later obtained a natural construction of a new chain (and cochain) complex $\overline{C}_{\bullet}(A,B,\varepsilon)$ (resp. $\overline{C}^{\bullet}(A,B,\varepsilon)$) in the process of introducing the secondary cyclic (co)homology. It turns out that unlike the classical case of associative algebras (over a field), there exist different (co)chain complexes for the $B$-algebra $A$. In this paper, we establish a connection between the two (co)homology theories for $B$-algebra $A$. We show that the pair $\big(H^{\bullet}((A,B,\varepsilon);A),HH_{\bullet}(A,B,\varepsilon)\big)$ forms a non-commutative differential calculus, where $HH_{\bullet}(A,B,\varepsilon)$ denotes the homology of the complex $\overline{C}_{\bullet}(A,B,\varepsilon)$., Comment: Any comments are welcome

A relative Rota-Baxter algebra is a triple $(A, M, T)$ consisting of an algebra $A$, an $A$-bimodule $M$, and a relative Rota-Baxter operator $T$. Using Voronov's derived bracket and a recent work of Lazarev et al., we construct an $L_\infty [1]$-algebra whose Maurer-Cartan elements are precisely relative Rota-Baxter algebras. By a standard twisting, we define a new $L_\infty [1]$-algebra that controls Maurer-Cartan deformations of a relative Rota-Baxter algebra $(A,M,T)$. We introduce the cohomology of a relative Rota-Baxter algebra $(A, M, T)$ and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of coboundary skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota-Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras., Comment: Version updated; 25 pages

$\mathcal{O}$-operators (also known as relative Rota-Baxter operators) on Lie algebras have several applications in integrable systems and the classical Yang-Baxter equations. In this article, we study $\mathcal{O}$-operators on hom-Lie algebras. We define cochain complex for $\mathcal{O}$-operators on hom-Lie algebras with respect to a representation. Any $\mathcal{O}$-operator induces a hom-pre-Lie algebra structure. We express the cochain complex of an $\mathcal{O}$-operator in terms of certain hom-Lie algebra cochain complex of the sub-adjacent hom-Lie algebra associated with the induced hom-pre-Lie algebra. If the structure maps in a hom-Lie algebra and its representation are invertible, then we can extend the above cochain complex to a deformation complex for $\mathcal{O}$-operators by adding the space of zero cochains. Subsequently, we study linear and formal deformations of $\mathcal{O}$-operators on hom-Lie algebras in terms of the deformation cohomology. In the end, we deduce deformations of $s$-Rota-Baxter operators (of weight 0) and skew-symmetric $r$-matrices on hom-Lie algebras as particular cases of $\mathcal{O}$-operators on hom-Lie algebras., Comment: Any suggestions or comments are welcome

In this article, we study the deformations of Filippov algebroids. We define a differential graded Lie algebra (in short DGLA) for a Filippov algebroid by introducing the notion of Filippov multiderivations for a vector bundle. Later on, we discuss deformations of a Filippov algebroid in terms of low-dimensional cohomology associated to this DGLA. We define Nijenhuis operators on Filippov algebroids and characterize trivial deformations of Filippov algebroids in terms of these operators. In the end, we define finite order deformations and discuss the problem of extending a given finite order deformation to a deformation of a higher order.

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra $(\mathfrak{gl}(V),[\cdot,\cdot],\mathsf{Ad})$, and the derivation Hom-Lie algebra of a Hom-Lie algebra.

We study formal deformations of hom-Lie-Rinehart algebras. The associated deformation cohomology that controls deformations is constructed using multiderivations of hom-Lie-Rinehart algebras., Comment: Any comments are welcome

We study deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base. This leads to describe the obstruction in extending a given deformation. We also discuss about the construction of versal deformation of Courant pairs. As an application, we explicitly compute universal infinitesimal deformation of Poisson algebra structures

In this paper we study universal central extensions and non-abelian tensor product of hom-Lie-Rinehart algebras. We discuss about universal $\alpha$- central extensions, and, lifting of automorphisms and $\alpha$-derivations to central extensions for hom-Lie-Rinehart algebras. This is in turn provide such lifting of automorphisms and $\alpha^k$-derivations to the central extensions for hom-Lie algebras., Comment: 22 pages

We define the notion of hom-Batalin-Vilkovisky algebras and strong differential hom-Gerstenhaber algebras as a special class of hom-Gerstenhaber algebras and provide canonical examples associated to some well-known hom-structures. Representations of a hom-Lie algebroid on a hom-bundle are defined and a cohomology of a regular hom-Lie algebroid with coefficients in a representation is studied. We discuss about relationship between these classes of hom-Gerstenhaber algebras and geometric structures on a vector bundle. As an application, we associate a homology to a regular hom-Lie algebroid and then define a hom-Poisson homology associated to a hom-Poisson manifold.

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism., Comment: Revised version, 23 pages

In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with coefficients in a module from the cochain complexes defining Hochschild cohomology and Leibniz cohomology., Comment: 23 pages, any comments are welcome