Articles written in Proceedings – Mathematical Sciences

    • Equivariant cobordism of Grassmann and flag manifolds

      Goutam Mukherjee

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      We consider certain natural (ℤ2)n actions on real Grassmann and flag manifolds andS1 actions on complex Grassmann manifolds with finite stationary point sets and determine completely which of them bound equivariantly.

    • Bredon cohomology of cyclic geometric realization ofG-cyclic sets

      Goutam Mukherjee Aniruddha C Naolekar

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      We define equivariant cyclic and Hochschild cohomology modules of a cyclic objectX in the category ofG-sets and relate them with the Bredon cohomologies of the cyclic geometric realization ¦X¦cy.

    • Nambu structures and associated bialgebroids


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      We investigate higher-order generalizations of well known results for Liealgebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu–Poisson manifold (of order $n$ > 2) gives rise to a special bialgebroid structure which is referred to as a weak Lie–Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu–Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie–Filippov bialgebroid over a point.

    • Nambu structures on Lie algebroids and their modular classes


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      We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $\Lambda$ as the modular class of the tangent Lie algebroid $TM$ with Nambu structure $\Lambda$. We show that many known properties of the modular class of a Nambu-Poisson manifold and that of a Lie algebroid extend to the setting of a Lie algebroid with Nambu structure. Finally, we prove that for a large class of Nambu-Poisson manifolds considered as tangent Lie algebroids with Nambu structures, the associated modular classes are closely related to Evens-Lu-Weinstein modular classes of Lie algebroids.

    • Cohomology and deformations of Filippov algebroids


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

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