Naive boundary strata and nilpotent orbits

Matt Kerr; Gregory Pearlstein

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Displaying similar documents to “Naive boundary strata and nilpotent orbits”

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...

$𝒟_{n, r}$ is not potentially nilpotent for $n \geq 4 r - 2$

Yan Ling Shao, Yubin Gao, Wei Gao (2016)

Czechoslovak Mathematical Journal

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An $n \times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$ . Let $𝒟_{n, r}$ be an $n \times n$ sign pattern with $2 \leq r \leq n$ such that the superdiagonal and the $(n, n)$ entries are positive, the $(i, 1)$ $(i = 1, \dots, r)$ and $(i, i - r + 1)$ $(i = r + 1, \dots, n)$ entries are negative, and zeros elsewhere. We prove that for $r \geq 3$ and $n \geq 4 r - 2$ , the sign pattern $𝒟_{n, r}$ is not potentially nilpotent, and so not spectrally arbitrary.

The orbits of the Hurwitz action of the braid groups on the standard generators

Yoshiro Yaguchi (2010)

Fundamenta Mathematicae

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The Hurwitz action of the n-braid group Bₙ on the n-fold direct product $(B_{m}) ⁿ$ of the m-braid group $B_{m}$ is studied. We show that the orbit of any n- tuple of the n standard generators of $B_{n + 1}$ consists of the (n-1)th powers of n+1 elements.

A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent

Fares Gherbi, Nadir Trabelsi (2024)

Czechoslovak Mathematical Journal

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Let $𝔐$ be the class of groups satisfying the minimal condition on normal subgroups and let $Ω$ be the class of groups of finite lower central depth, that is groups $G$ such that $γ_{i} (G) = γ_{i + 1} (G)$ for some positive integer $i$ . The main result states that if $G$ is a finitely generated hyper-(Abelian-by-finite) group such that for every $x \in G$ , there exists a normal subgroup $H_{x}$ of finite index in $G$ satisfying $〈 x, x^{h} 〉 \in 𝔐 Ω$ for every $h \in H_{x}$ , then $G$ is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated...

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with $p_{g} = 0$ obtained as $0$ -sets $X_{σ}$ of a section $σ$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$ -forms of $X_{σ}$ , then it acts as $0$ on $0$ -cycles of degree $0$ of $X_{σ}$ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{σ}$ ...

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...

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...

Retracts that are kernels of locally nilpotent derivations

Dayan Liu, Xiaosong Sun (2022)

Czechoslovak Mathematical Journal

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Let $k$ be a field of characteristic zero and $B$ a $k$ -domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$ . We show that if $B = R \oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B = R^{[1]}$ , i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$ -UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$ , then does it follow that $B ≅ R^{[2]}$ ? We give a negative answer to this question. The interest in...

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...

Characterization of cycle domains via Kobayashi hyperbolicity

Gregor Fels, Alan Huckleberry (2005)

Bulletin de la Société Mathématique de France

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A real form $G$ of a complex semi-simple Lie group $G^{ℂ}$ has only finitely many orbits in any given $G^{ℂ}$ -flag manifold $Z = G^{ℂ} / Q$ . The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle...

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space $Λ^{p}$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^{p}$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that $Λ^{p}$ splits as the direct sum $d * (Λ^{p + 1}) \oplus δ * (Λ^{p - 1}) \oplus H^{p}$ , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

A new family of spectrally arbitrary ray patterns

Yinzhen Mei, Yubin Gao, Yan Ling Shao, Peng Wang (2016)

Czechoslovak Mathematical Journal

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An $n \times n$ ray pattern $𝒜$ is called a spectrally arbitrary ray pattern if the complex matrices in $Q (𝒜)$ give rise to all possible complex polynomials of degree $n$ . In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an $n \times n$ irreducible spectrally arbitrary ray pattern is $3 n - 1$ . In this paper, we introduce a new family of spectrally arbitrary ray patterns of order $n$ with exactly $3 n - 1$ nonzeros.

A note on infinite $a S$ -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $a S$ -group. We study some properties of $a S$ -groups. For instance, it is shown that a nilpotent group $G$ is an $a S$ -group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $a S$ -group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

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)...