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Pralay Chatterjee

Publications and source records attributed to Pralay Chatterjee.

8 recordsLinked to original sources
arXiv2023

On the third and fourth Betti numbers of a homogeneous space of a Lie group

In the paper "The second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee, and C. Maity, explicit descriptions of the second and first real de Rham cohomology groups of a general homogeneous space of a Lie group are given, extending an earlier result in "On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits, Internat. J. Math. 23 (2012), no. 8, 1250086" by I. Biswas and P. Chatterjee. From the computational viewpoint, they turned out to be new and very useful, and in fact played a crucial role in determining the second cohomology of nilpotent orbits as done in the above two papers. In this paper, we give computable and explicit descriptions of the third and fourth real de Rham cohomologies of a general homogeneous space, in terms of the associated Lie-theoretic data, along the lines mentioned above. We also draw numerous corollaries of our main results in important special settings. Moreover, as a consequence, we obtain a new and interesting invariant by showing that for a large class of homogeneous spaces, the difference between the third and fourth Betti numbers coincides with the difference between the numbers of simple factors of the ambient group and the associated closed subgroup.

math.GR↗
arXiv2022

Homotopy type of the nilpotent orbits in classical Lie algebras

In the paper "The Second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee and C. Maity homotopy types of nilpotent orbits are explicitly described in the case of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is not semisimple. In this paper we extend the above description of homotopy type of nilpotent orbits to the remaining cases of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is semisimple.

math.GR↗
arXiv2016

The second cohomology groups of nilpotent orbits in classical Lie algebras

The second de Rham cohomology groups of nilpotent orbits in non-compact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in non-compact classical simple Lie algebras are computed; they are proven to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes a previous theorem of the first two authors, may be of independent interest.

math.GR↗
arXiv2016

On the second cohomology of nilpotent orbits in exceptional Lie algebras

In this paper we consider non-compact non-complex exceptional Lie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotent orbits. For the rest of cases of nilpotent orbits, which are not covered in the above computations, we obtain upper bounds for the dimensions of the second cohomology groups.

math.GR↗
arXiv2016

On abstract homomorphisms of algebraic groups

In this paper we study abstract group homomorphisms between the groups of rational points of linear algebraic groups which are not necessarily reductive. One of our main goal is to obtain results on homomorphisms from the groups of rational points of linear algebraic groups defined over certain specific fields to the groups of rational points of linear algebraic groups over number fields and non-archimedean local fields of characteristic zero; in this set-up we deal with the unexplored topic of abstract homomorphisms from the groups of rational points of anisotropic groups over non-archimedean local fields of characteristic zero. We also obtain results on abstract homomorphisms from unipotent and solvable groups, and prove general results on the structures of abstract homomorphisms using the celebrated result of Borel and Tits in this area and a well-known theorem due to Tits on the structure of the groups of rational points of isotropic semisimple groups.

math.GR↗
arXiv2004

Divergent torus orbits in homogeneous spaces of Q-rank two

Let G be the real points of a semisimple algebraic Q-group, let H be an arithmetic subgroup of G and let T be the real points of an R-split torus in G. We prove that if there is a divergent T-orbit in G/H, and Q-rank(G) > 1, then the dimension of T is not larger than Q-rank(G). This provides a partial answer to a question of G.Tomanov and B.Weiss.

math.DG↗