Indagationes Mathematicae-New Series最新文献

We study Dirac cohomology $H_D^{g,h}\left(M\right)$ for modules belonging to category $O$ of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of $M\otimes S$, with $S$ being a spin module of $h^{\perp }$. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for $H_D^{g,h}\left(M\right)$ while we show that in the case of a Hermitian symmetric pair $(g,k)$ and an irreducible unitary module $M\in O$, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in $M$. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for $M\in O$.

In this article, we define the tensor product $V\otimes W$ of a representation $V$ of a quiver $Q$ with a representation $W$ of an another quiver $Q^{\prime }$, and show that the representation $V\otimes W$ is semistable if $V$ and $W$ are semistable. We give a relation between the universal representations on the fine moduli spaces $N_1,N_2$ and $N_3$ of representations of $Q,Q^{\prime }$ and $Q\otimes Q^{\prime }$ respectively over arbitrary algebraically closed fields. We further describe a relation between the natural line bundles on these moduli spaces when the base is the field of complex numbers. We then prove that the internal product $\tilde{Q}\otimes \widetilde{Q^{\prime }}$ of covering quivers is a sub-quiver of the covering quiver $\widetilde{Q\otimes Q^{\prime }}$. We deduce the relation between stability of the representations $\widetilde{V\otimes W}$ and $\tilde{V}\otimes \tilde{W}$, where $\tilde{V}$ denotes the lift of the representation $V$ of $Q$ to the covering quiver $\tilde{Q}$. We also lift the relation between the natural line bundles on the product of moduli spaces $\widetilde{N_1}\times \widetilde{N_2}$.

We prove a normal form theorem for principal Hamiltonian actions on Poisson manifolds around the zero locus of the moment map. The local model is the generalization to Poisson geometry of the classical minimal coupling construction from symplectic geometry of Sternberg and Weinstein. Further, we show that the result implies that the quotient Poisson manifold is linearizable, and we show how to extend the normal form to other values of the moment map.

For integer $n$ and real $u$, define $\Delta (n,u)≔\left|\{d:d\mid n,e^u<d⩽e^{u+1}\}\right|$. Then, the Erdős–Hooley Delta function is defined as $\Delta \left(n\right)≔{max}_{u\in R}\Delta (n,u).$ We provide uniform upper and lower bounds for the mean-value of $\Delta \left(n\right)$ over friable integers, i.e. integers free of large prime factors.

In this paper, among other things, we explicit a $G_{\delta }$-dense set of Liouville numbers, for which the triple power tower of any of its elements is a transcendental number.

The aim of this note is twofold. Firstly, we explain in detail Remark 4.1 in Doan (2020) by showing that the action of the automorphism group of the second Hirzebruch surface $F_2$ on itself extends to its formal semi-universal deformation only up to the first order. Secondly, we show that for reductive group actions, the locality of the extended actions on the Kuranishi space constructed in Doan (2021) is the best one could expect in general.

In this survey, we discuss the definition of a (quasi-)Banach function space. We advertise the original definition by Zaanen and Luxemburg, which does not have various issues introduced by other, subsequent definitions. Moreover, we prove versions of well-known basic properties of Banach function spaces in the setting of quasi-Banach function spaces.

For $m$ an even positive integer and $p$ an odd prime, we show that the generalized Euler polynomial $E_{mp}^{\left(mp\right)}\left(x\right)$ is in Eisenstein form with respect to $p$ if and only if $p$ does not divide $m(2^m-1)B_m$. As a consequence, we deduce that at least $1/3$ of the generalized Euler polynomials $E_n^{\left(n\right)}\left(x\right)$ are in Eisenstein form with respect to a prime $p$ dividing $n$ and, hence, irreducible over $Q$.

The integral cohomology algebra of ${\tilde{G}}_{6,3}$ has been determined in the recent work of Kalafat and Yalçınkaya. We completely determine the integral cohomology algebra of ${\tilde{G}}_{n,3}$ for $n=8$ and $n=10$. The main method used to describe these algebras is the Leray–Serre spectral sequence. We also illustrate this method by determining the integral cohomology algebra of ${\tilde{G}}_{n,2}$ for $n$ odd.

We reduce the Mathieu conjecture for $SU\left(2\right)$ to a conjecture about moments of Laurent polynomials in two variables with single variable polynomial coefficients.