Poisson-Nijenhuis groupoids

We define multiplicative Poisson-Nijenhuis structures on a Lie groupoid which extends the notion of symplectic-Nijenhuis groupoid introduced by Sti\'e23non and Xu. We also introduce a special class of Lie bialgebroid structure on a Lie algebroid $A$, called P-N Lie bialgebroid, which defines a hierarchy of compatible Lie bialgebroid structures on $A$. We show that under some topological assumption on the groupoid, there is a one-to-one correspondence between multiplicative Poisson-Nijenhuis structures on a Lie groupoid and P-N Lie bialgebroid structures on the corresponding Lie algebroid.Comment: 23 page

Deformations of Loday-type algebras and their morphisms

We study formal deformations of multiplication in an operad. This closely resembles Gerstenhaber's deformation theory for associative algebras. However, this applies to various algebras of Loday-type and their twisted analogs. We explicitly describe the cohomology of these algebras with coefficients in a representation. Finally, deformation of morphisms between algebras of the same Loday-type is also considered.Comment: This is the final version that appears in "Journal of Pure and Applied Algebra

Contact Courant algebroids and L∞L_\infty-algebras

Let $L$ be a line bundle over $M$. In this paper we associate an $L_\infty$-algebra to any $L$-Courant algebroid (contact Courant algebroid in the sense of Grabowski). This construction is similar to the work of Roytenberg and Weinstein for Courant algebroids. Next we associate a $p$-term $L_\infty$-algebra to any isotropic involutive subbundle of $(\mathbb{D}L)^p := DL \oplus (\wedge^p (DL)^* \otimes L)$, where $DL$ is the gauge algebroid of $L$. In a particular case, we relate these $L_\infty$-algebras by a suitable morphism.Comment: 19 pages, comments are welcom

Gerstenhaber algebra structure on the cohomology of a hom-associative algebra

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we define a cup product on the cohomology of a hom-associative algebra. We show that the cup product together with the degree $-1$ graded Lie bracket (which controls the deformation of the hom-associative algebra structure) on the cohomology forms a Gerstenhaber algebra. This generalizes a classical fact that the Hochschild cohomology of an associative algebra carries a Gerstenhaber algebra structure.Comment: 14 pages; final version; accepted for publication in Proc. Indian Acad. Sci. Math. Sci.; comments are still welcom

Deformations of associative Rota-Baxter operators

Rota-Baxter operators and more generally $\mathcal{O}$-operators on associative algebras are important in probability, combinatorics, associative Yang-Baxter equation and splitting of algebras. Using a method of Uchino, we construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by $\mathcal{O}$-operators. This allows us to construct a cohomology for an $\mathcal{O}$-operator. This cohomology can also be seen as the Hochschild cohomology of a certain algebra with coefficients in a suitable representation. Next, we study linear and formal deformations of an $\mathcal{O}$-operator which are governed by the above-defined cohomology. We introduce Nijenhuis elements associated with an $\mathcal{O}$-operator which give rise to trivial deformations. As an application, we conclude deformations of weight zero Rota-Baxter operators and associative {\bf r}-matrices.Comment: Comments are welcome; final version appear in Journal of Algebr

On Generalized Lie Bialgebroids

An alternative proof of the duality of generalized Lie bialgebroid is given and proved a canonical Jacobi structure can be defined on the base of it. We also introduce the notion of morphism between generalized Lie bialgebroids and proved that the induced Jacobi structure is unique upto a morphism.Comment: 15 page

Crossed extensions of Lie algebras

It is known that Hochschild cohomology groups are represented by crossed extensions of associative algebras. In this paper, we introduce crossed $n$-fold extensions of a Lie algebra $\mathfrak{g}$ by a module $M$, for $n \geq 2$. The equivalence classes of such extensions are represented by the $(n+1)$-th Chevalley-Eilenberg cohomology group $H^{n+1}_{CE} (\mathfrak{g}, M).$Comment: Comments are welcom

Cohomology of BiHom-associative algebras

Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.Comment: This is the final version that appears in "Journal of Algebra and its Applications

Hom-associative algebras up to homotopy

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we introduce a strongly homotopy version of hom-associative algebras ($HA_\infty$-algebras in short) on a graded vector space. We describe $2$-term $HA_\infty$-algebras in details. In particular, we study 'skeletal' and 'strict' $2$-term $HA_\infty$-algebras. We also introduce hom-associative $2$-algebras as categorification of hom-associative algebras. The category of $2$-term $HA_\infty$-algebras and the category of hom-associative $2$-algebras are shown to be equivalent. An appropriate skew-symmetrization of $HA_\infty$-algebras give rise to $HL_\infty$-algebras introduced by Sheng and Chen. Finally, we define a suitable Hochschild cohomology theory for $HA_\infty$-algebras which control the deformation of the structures.Comment: 30 pages, comments are welcom

Kink instability of a highly deformable elastic cylinder

When a soft elastic cylinder is bent beyond a critical radius of curvature, a sharp fold in the form of a kink appears at its inner side while the outer side remains smooth. The critical radius increases linearly with the diameter of the cylinder while remaining independent of its elastic modulus, although, its maximum deflection at the location of the kink depends on both the diameter and the modulus of the cylinders. Experiments are done also with annular cylinders of varying wall thickness which exhibits both the kinking and the ovalization of the cross-section. The kinking phenomenon appears to occur by extreme localization of curvature at the inner side of a post-buckled cylinder.Comment: 7 pages, 4 figure