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Kentaro Mikami

Publications and source records attributed to Kentaro Mikami.

15 recordsLinked to original sources
arXiv2025

Super homology groups of differential forms and vector fields on Euclidean line

The arXiv:2105.09738 claims several stuffs. In particular, we recall the following two. (1) Vector fields and differential forms become a Lie superalgebra structure for each manifold. (2) For an n-dimensional Euclidean space, vector fields and differential forms with polynomial coefficients become a double weighted Lie uperalgebra. By using Euler vector field, the Betti numbers are 0 except the last one if the primary weight and the secondary weight are different. Now, a simple question arises: What happens when the primary weight and the secondary weight are equal? This note shall give a complete answer to the question for the case $n=1$.

math.DG↗
arXiv2023

Poisson-like cohomologies associated with some Lie superalgebras

The concept of Poisson cohomology groups associated with Poisson manifolds is a part of the theory of Lie superalgebras of vector fields. Therefore, we abstracted them as Poisson-like cohomology groups for general Lie superalgebras. In doing so, we obtained a generalization of the concept of the Euler number. For the Lie superalgebras of differential forms on a manifold, we found the de Rham cohomology groups match with the Poisson-like cohomology groups in the special case. In order to understand the development of the discussion, we presented some simple examples and show ideas of how our discussion unfolds.

math.DG↗
arXiv2021

Deformed super brackets on forms of a manifold

Given a manifold, we have a super bracket on the graded algebra of differential forms by {A,B} = (-1)^{a} d(A \wedge B). We study when {A,B}_{t} = (-1)^{a} d(A \wedge B) + F(a,b) A \wedge t ϕ\wedge B becomes super bracket for a 1-form ϕ. And we show a concrete small example where the situation of parameter t is not identical.

math.SG↗
arXiv2021

Superalgebra structure on differential forms of manifold

As an analogy of superalgebra of multivector fields with the Schounte bracket, we introduce a non-trivial superbracket on differential forms of manifold. We show properties of this new superalgebra. We extend this superalgebra by adding one factor. The new extended superalgebra should be studied more widely and in deep. We study Betti numbers of double weighted homology groups by the Euler vector field. In appendix, we explain our bracket is produced like as the Schouten bracket.

math.GM↗
arXiv2020

Super homologies associated with low dimensional Lie algebras

A Poisson structure on a manifold is characterized by the Schouten bracket. The graded algebra of the tangent bundle with the Schouten bracket is a prototype of Lie superalgebra. The Poisson condition means that a cycle in the 2-chain space. Given a graded Lie superalgebra, the 0-graded subspace is a Lie algebra. In this note, using the DGA of tangent bundle with the Schouten bracket as a model, we start from an abstract Lie algebra, construct non-trivial Lie superalgebra by Schouten-like bracket. Then it is natural to ask how the core Lie algebra control the Lie superalgebra. One trial here is to investigate the Betti numbers of the super homology groups. For abelian Lie algebras, the boundary operator is trivial, so we study super homology groups for low dimensional non-abelian Lie algebras of dimension smaller than 4.

math.DG↗
arXiv2018

Euler number and Betti numbers of homology groups of pre Lie superalgebra

First we recall homology groups of prer Lie superalgebras. Then introducing double weighted chain spaces, we deal with pre Lie superalgebra of multi-vector fields with polynomial coefficients on n-dimensional number space. The bracket is Schouten bracket. We have several results about Euler number and Betti numbers of those homology groups of special pre Lie superalgebra.

math.DG↗
arXiv2017

Cohomology groups of homogeneous Poisson structures

We generalize the notion of weight for Gelfan'd-Fuks cohomology theory of symplectic vector spaces to the homogeneous Poisson vector spaces, and try some combinatorial approach to Poisson cohomology groups.

math.SG↗
arXiv2014

An affirmative answer to a conjecture for Metoki class

In "The {G}el'fand-{K}alinin-{F}uks class and characteristic classes of transversely symplectic foliations" arXiv:0910.3414, Kotschick and Morita showed that the Gel'fand-Kalinin-Fuks class in $\ds\HGF{7}{2}{}{8}$ is decomposed as a product $η\wedge ω$ of some leaf cohomology class $η$ and a transverse symplectic class $ω$. We show that the same formula holds for Metoki class, which is a non-trivial element in $\ds \HGF{9}{2}{}{14}$. The result was conjectured by Kotschick and Morita, where they studied characteristic classes of symplectic foliations due to Kontsevich. Our proof depends on Groebner Basis theory using computer calculations.

math.DG↗
arXiv2014

Another proof to Kotschick-Morita's Theorem of Kontsevich homomorphism

In \cite{KOT:MORITA}, Kotschick and Morita showed that the Gel'fand-Kalinin-Fuks class in $\ds \HGF{7}{2}{}{8}$ is decomposed as a product $η\wedge ω$ of some leaf cohomology class $η$ and a transverse symplectic class $ω$. In other words, the Kontsevich homomorphism $\dsω\wedge :\HGF{5}{2}{0}{10} \rightarrow\HGF{7}{2}{}{8}$ is isomorphic. In this paper, we give proof for the Kotschick and Morita's theorem by using the Gröbner Basis theory and computer symbol calculations.

math.DG↗
arXiv2014

Lower weight Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4

In this paper, we investigate the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4. In the case of formal Hamiltonian vector fields on R^2, we computed the relative Gel'fand-Kalinin-Fuks cohomology groups of weight <20 in the paper by Mikami-Nakae-Kodama. The main strategy there was decomposing the Gel'fand-Fucks cochain complex into irreducible factors and picking up the trivial representations and their concrete bases, and ours is essentially the same. By computer calculation, we determine the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4 of weights 2, 4 and 6. In the case of weight 2, the Betti number of the cohomology group is equal to 1 at degree 2 and is 0 at any other degree. In weight 4, the Betti number is 2 at degree 4 and is 0 at any other degree, and in weight 6, the Betti number is 0 at any degree.

math.AT↗
arXiv2014

Higher weight Gel'fand-Kalinin-Fuks classes of formal Hamiltonian vector fields of symplectic R^2

In "The Gel'fand-Kalinin-Fuks class and characteristic classes of transversely symplectic foliations", arXiv:0910.3414, (October 2009) by D.Kotschick and S.Morita, the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields without constant vector fields on 2n-plane were characterized by two parameters, one is degree and the other is weight. And they obtained those cohomology groups of the 2-plane while their weight <= 10. In this paper, for those cohomology groups of the 2-plane, we succeeded in determining the dimension of cochain complexes by Sp(2,R)-representation theory for their weight even less than 50, thus, we manipulate the Euler characteristic numbers. We also decide our relative Gel'fand-Kalinin-Fuks cohomology groups until whose weight < 20 by getting a concrete matrix representation of the coboundary operator.

math.SG↗