Invariants of complex structures on nilmanifolds

Edwin Alejandro Rodríguez Valencia

Displaying similar documents to “Invariants of complex structures on nilmanifolds”

Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center

Bin Ren, Lin Sheng Zhu (2017)

Czechoslovak Mathematical Journal

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A Lie algebra $L$ is called 2-step nilpotent if $L$ is not abelian and $[L, L]$ lies in the center of $L$ . 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.

Parallelepipeds, nilpotent groups and Gowers norms

Bernard Host, Bryna Kra (2008)

Bulletin de la Société Mathématique de France

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In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions $2$ and $3$ and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.

Leibniz's rule on two-step nilpotent Lie groups

Krystian Bekała (2016)

Colloquium Mathematicae

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Let be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows us to define a generalized multiplication $f g = {(f^{\lor} * g^{\lor})}^{\land}$ of two functions in the Schwartz class (*), where $^{\lor}$ and $^{\land}$ are the Abelian Fourier transforms on the Lie algebra and on the dual * and ∗ is the convolution on the group . In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order...

Sub-Laplacian with drift in nilpotent Lie groups

Camillo Melzi (2003)

Colloquium Mathematicae

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We consider the heat kernel $ϕ_{t}$ corresponding to the left invariant sub-Laplacian with drift term in the first commutator of the Lie algebra, on a nilpotent Lie group. We improve the results obtained by G. Alexopoulos in [1], [2] proving the “exact Gaussian factor” exp(-|g|²/4(1+ε)t) in the large time upper Gaussian estimate for $ϕ_{t}$ . We also obtain a large time lower Gaussian estimate for $ϕ_{t}$ .

Asymptotic behavior of the invariant measure for a diffusion related to an NA group

Ewa Damek, Andrzej Hulanicki (2006)

Colloquium Mathematicae

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On a Lie group NA that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, the heat semigroup $μ_{t}$ generated by a second order subelliptic left-invariant operator $\sum_{j = 0}^{m} Y_{j} + Y$ is considered. Under natural conditions there is a $μ ̌_{t}$ -invariant measure m on N, i.e. $μ ̌_{t} * m = m$ . Precise asymptotics of m at infinity is given for a large class of operators with Y₀,...,Yₘ generating the Lie algebra of S.

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

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Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to $ℝ^{k}$ , k>1. We consider a class of second order left-invariant differential operators on S of the form $ℒ_{α} = L^{a} + Δ_{α}$ , where $α \in ℝ^{k}$ , and for each $a \in ℝ^{k}, L^{a}$ is left-invariant second order differential operator on N and $Δ_{α} = Δ - ⟨ α, \nabla ⟩$ , where Δ is the usual Laplacian on $ℝ^{k}$ . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively)...

Geodesically equivalent metrics on homogenous spaces

Neda Bokan, Tijana Šukilović, Srdjan Vukmirović (2019)

Czechoslovak Mathematical Journal

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Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two $G$ -invariant metrics of arbitrary signature on homogenous space $G / H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, $G$ -invariant metrics on homogenous space $G / H$ implies that their holonomy algebra cannot be full. We give an algorithm...

Convergence of formal solutions of first order singular partial differential equations of nilpotent type

Masatake Miyake, Akira Shirai (2012)

Banach Center Publications

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Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $P u (x, y, z) : = (y \partial_{x} - z \partial_{y}) u (x, y, z) = f (x, y, z) \in_{x, y, z}$ , where $P = y \partial_{x} - z \partial_{y} :_{x, y, z} \to_{x, y, z}$ . For this equation, our aim is to characterize the solvability on $_{x, y, z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.

Hurewicz-Serre theorem in extension theory

M. Cencelj, J. Dydak, A. Mitra, A. Vavpetič (2008)

Fundamenta Mathematicae

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The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that $X τ K (H_{k} (L), k)$ for all k ≥ 1, then $X τ K (π_{k} (F), k)$ and $X τ K (π_{k} (L), k)$ for all k ≥ $.$ Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈...

On the Finsler geometry of the Heisenberg group $H_{2 n + 1}$ and its extension

Mehri Nasehi (2021)

Archivum Mathematicum

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We first classify left invariant Douglas $(α, β)$ -metrics on the Heisenberg group $H_{2 n + 1}$ of dimension $2 n + 1$ and its extension i.e., oscillator group. Then we explicitly give the flag curvature formulas and geodesic vectors for these spaces, when equipped with these metrics. We also explicitly obtain $S$ -curvature formulas of left invariant Randers metrics of Douglas type on these spaces and obtain a comparison on geometry of these spaces, when equipped with left invariant Douglas $(α, β)$ -metrics. More exactly,...

Partial differential equations in Banach spaces involving nilpotent linear operators

Antonia Chinnì, Paolo Cubiotti (1996)

Annales Polonici Mathematici

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Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_{t}^{k} u + \sum_{j = 0}^{k - 1} \sum_{| α | \leq m} A_{j, α} (D_{t}^{j} D_{x}^{α} u) = f$ in $ℝ^{n + 1}$ , ⎨ ⎩ $D_{t}^{j} u (0, x) = φ_{j} (x)$ in $ℝ^{n}$ , j=0,...,k-1, where each $A_{j, α}$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j, α}$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u \in C^{\infty} (ℝ^{n + 1}, E)$ whose derivatives are equi-bounded on each bounded subset of $ℝ^{n + 1}$ .

On the nilpotent residuals of all subalgebras of Lie algebras

Wei Meng, Hailou Yao (2018)

Czechoslovak Mathematical Journal

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Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$ , there exists a smallest ideal $I$ of $L$ such that $L / I \in 𝒩$ . This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{𝒩}$ . In this paper, we define the subalgebra $S (L) = ⋂_{H \leq L} I_{L} (H^{𝒩})$ . Set $S_{0} (L) = 0$ . Define $S_{i + 1} (L) / S_{i} (L) = S (L / S_{i} (L))$ for $i \geq 1$ . By $S_{\infty} (L)$ denote the terminal term of the ascending series. It is proved that $L = S_{\infty} (L)$ if and only if $L^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a...

Images of locally nilpotent derivations of bivariate polynomial algebras over a domain

Xiaosong Sun, Beini Wang (2024)

Czechoslovak Mathematical Journal

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We study the LND conjecture concerning the images of locally nilpotent derivations, which arose from the Jacobian conjecture. Let $R$ be a domain containing a field of characteristic zero. We prove that, when $R$ is a one-dimensional unique factorization domain, the image of any locally nilpotent $R$ -derivation of the bivariate polynomial algebra $R [x, y]$ is a Mathieu-Zhao subspace. Moreover, we prove that, when $R$ is a Dedekind domain, the image of a locally nilpotent $R$ -derivation of $R [x, y]$ with some...

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

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We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜 = 𝕂 [𝔸^{n}]$ is said to be of Kostant type, if its centre $Z (𝒜)$ is freely generated by homogeneous polynomials $F_{1}, ..., F_{r}$ such that they give Kostant’s regularity criterion on $𝔸^{n} (d_{x} F_{i}$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$ ). If the initial Poisson algebra is of Kostant type and $F_{i}$ satisfy a certain degree-equality, then the contraction...

On some properties of the upper central series in Leibniz algebras

Leonid A. Kurdachenko, Javier Otal, Igor Ya. Subbotin (2019)

Commentationes Mathematicae Universitatis Carolinae

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This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra $L$ includes a finite dimensional ideal $K$ such that the factor-algebra $L / K$ is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.