Journal of Mathematics of Kyoto University最新文献

英文中文

A Gauss-Bonnet-type formula on Riemann-Finsler surfaces with nonconstant indicatrix volume 非常指标体积Riemann-Finsler曲面上的gauss - bonnet型公式

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2010-03-01 DOI: 10.1215/0023608X-2009-008

J. Itoh, S. Sabau, H. Shimada

We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.

证明了具有规则分段光滑边界的非常指标体积Riemann-Finsler曲面的Gauss-Bonnet型公式。给出了兰兹伯曲面n个平行线的Hadamard型定理。

引用次数: 7

Quantum continuous $mathfrak{gl}_{infty}$: Semiinfinite construction of representations 量子连续$mathfrak{gl}_{infty}$:表示的半无限构造

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2010-02-16 DOI: 10.1215/21562261-1214375

B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin

We begin a study of the representation theory of quantum continuous $mathfrak{gl}_infty$, which we denote by $mathcal E$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of $mathcal E$ are labeled by a continuous parameter $uin {mathbb C}$. The representation theory of $mathcal E$ has many properties familiar from the representation theory of $mathfrak{gl}_infty$: vector representations, Fock modules, semi-infinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from $mathcal E$ to spherical double affine Hecke algebras $Sddot H_N$ for all $N$. A key step in this construction is an identification of a natural bases of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the $K$-theory of Hilbert schemes.

我们开始研究量子连续的表示理论$mathfrak{gl}_infty$，我们用$mathcal E$表示。该代数依赖于两个参数，是作用于一元洛朗多项式空间的差分算子李代数的包络代数的变形版本。$mathcal E$的基本表示用一个连续参数$uin {mathbb C}$来标记。$mathcal E$的表示理论有许多与$mathfrak{gl}_infty$的表示理论相似的性质:向量表示、Fock模块、模块的半无限构造。利用向量表示的张量积，构造了从$mathcal E$到所有$N$的球面双仿射Hecke代数$Sddot H_N$的满射同态。这个构造的关键步骤是确定向量表示与麦克唐纳多项式的张量积的自然基。我们还证明了其中一个Fock表示与先前利用Hilbert格式的$K$ -理论构造的模块是同构的。

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引用次数: 86

Quantum continuous $mathfrak{gl}_{infty}$: Tensor products of Fock modules and $mathcal{W}_{n}$-characters 量子连续$mathfrak{gl}_{infty}$: Fock模的张量积与$mathcal{W}_{n}$ -字符

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2010-02-16 DOI: 10.1215/21562261-1214384

B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin

We construct a family of irreducible representations of the quantum continuous $gl_infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type. In particular, we obtain a simple combinatorial model for all representations of the $W_n$-algebras appearing in the minimal models in terms of $n$ interrelating partitions.

构造了量子连续$gl_infty$的一组不可约表示，这些表示的性质与$gl_n$型$W_n$代数的最小模型中的表示的性质一致。特别地，我们获得了一个简单的组合模型，用于最小模型中以$n$相互关联分区表示的$W_n$ -代数的所有表示。

引用次数: 52

Gorenstein flat dimension of complexes 戈伦斯坦复合体的平面维数

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2010-02-09 DOI: 10.1215/KJM/1265899484

A. Iacob

We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein coho- mology for complexes; we also define a generalized Tate cohomol- ogy for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.

定义了左gf闭环上无界配合物的Gorenstein平面维数的概念。在Gorenstein环上，我们引入了一个关于配合物的Gorenstein同模的概念;我们还定义了Gorenstein环上配合物的广义Tate上同调，并证明了绝对上同调、Gorenstein上同调和广义Tate上同调之间存在着密切的联系。

引用次数: 34

The Fano surface of the Klein cubic threefold 克莱因三次立方的范诺曲面

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2010-01-27 DOI: 10.1215/KJM/1248983032

X. Roulleau

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order $11$. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index $2$ sub-group of the Neron-Severi group and we obtain a set of generators of this group. The Neron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.

证明了克莱因三次元$F$是唯一具有11阶自同构的光滑三次元$F$。我们计算了F的中间雅可比矩阵的周期格，并研究了它的Fano曲面。我们还计算了$S$在正属曲线上的纤维集和这些纤维之间的交点。这些纤维产生了Neron-Severi群的index $2$子群，我们得到了该群的一组发生器。$S$的Neron-Severi群的秩$25=h^{1,1}$和判别式$11^{10}$。

引用次数: 19

On the WKB-theoretic structure of a Schrödinger operator with a merging pair of a simple pole and a simple turning point 简单极点与简单拐点合并对Schrödinger算子的wkb理论结构

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2010-01-01 DOI: 10.1215/0023608X-2009-007

S. Kamimoto, T. Kawai, T. Koike, Yoshitsugu Takei

A Schrödinger equation with a merging pair of a simple pole and a simple turning point (called MPPT equation for short) is studied from the viewpoint of exact Wentzel-Kramers-Brillouin (WKB) analysis. In a way parallel to the case of mergingturning-points (MTP) equations, we construct a WKB-theoretic transformation that brings anMPPTequation to its canonical form (the ∞-Whittaker equation in this case). Combining this transformation with the explicit description of the Voros coefficient for the Whittaker equation in terms of the Bernoulli numbers found by Koike, we discuss analytic properties of Borel-transformed WKB solutions of an MPPT equation. 0. Introduction The principal aim of this article is to form a basis for the exact WKB analysis of a Schrödinger equation (0.1) ( d dx2 − ηQ(x, η) ) ψ = 0 (η: a large parameter) with one simple turning point and with one simple pole in the potential Q. As [Ko1] and [Ko3] emphasize, the Borel transform of a WKB solution of (0.1) displays, near the simple pole singularity, behavior similar to that near a simple turning point. Hence it is natural to expect that such an equation plays an important role in exact WKB analysis in the large. Such an expectation has recently been enhanced by the discovery (see [KoT]) that the Voros coefficient of a WKB solution of (0.1) with (0.2) Q = 1 4 + α x + η−2 γ x2 (α, γ: fixed complex numbers) can be explicitly written down with the help of the Bernoulli numbers. The potential Q given by (0.2) plays an important role in Section 2; the Schrödinger Kyoto Journal of Mathematics, Vol. 50, No. 1 (2010), 101–164 DOI 10.1215/0023608X-2009-007, © 2010 by Kyoto University Received July 30, 2009. Revised October 2, 2009. Accepted October 9, 2009. Mathematics Subject Classification: Primary 34M60; Secondary 34E20, 34M35, 35A27, 35A30. Authors’ research supported in part by Japan Society for the Promotion of Science Grants-in-Aid 20340028, 21740098, and 21340029. 102 Kamimoto, Kawai, Koike, and Takei equation with the potential Q of the form (0.2), that is, the Whittaker equation with a large parameter η, gives us a WKB-theoretic canonical form of a Schrödinger equation with one simple turning point and with one simple pole in its potential. We note that the parameter α contained in the Whittaker equation in Section 2 is an infinite series α(η) = ∑ k≥0 αkη −k (αk: a constant), and we call such an equation the ∞-Whittaker equation when we want to emphasize that α is not a genuine constant but an infinite series as above. In order to make a semiglobal study of a Schrödinger equation with one simple turning point and with a simple pole in its potential, we let the simple pole singular point merge with the turning point and observe what kind of equation appears. For example, what if we let α tend to zero in (0.2) with γ being kept intact? Interestingly enough, the resulting equation is what we call a ghost equation (see [Ko2]); we have been wondering where we should place the class of g

本文从精确WKB分析的角度研究了具有简单极点和简单拐点合并对的Schrödinger方程(简称MPPT方程)。以一种与合并拐点(MTP)方程类似的方式，我们构造了一个wkb理论变换，将mpp方程转化为其标准形式(在这种情况下为∞-Whittaker方程)。结合这种变换和用Koike发现的伯努利数对Whittaker方程的Voros系数的显式描述，讨论了MPPT方程borel变换WKB解的解析性质。0. 本文的主要目的是形成一个精确的WKB分析的基础Schrödinger方程(0.1)(d dx2−η q (x， η)) ψ = 0 (η:一个大参数)具有一个简单的拐点和一个简单的极位q。正如[Ko1]和[Ko3]所强调的，(0.1)的WKB解的Borel变换在简单极奇点附近表现出类似于在简单拐点附近的行为。因此，很自然地期望这样的方程在大规模的精确WKB分析中起重要作用。这种期望最近被一项发现(见[KoT])所加强，即(0.2)Q = 14 + α x + η−2 γ x2 (α， γ:固定复数)的(0.1)WKB解的Voros系数可以在伯努利数的帮助下明确地写出来。由(0.2)给出的电位Q在第2节中起重要作用;Schrödinger京都数学杂志，Vol. 50, No. 1 (2010)， 101-164 DOI 10.1215/0023608X-2009-007，©2010 by Kyoto University，收于2009年7月30日。2009年10月2日修订。2009年10月9日录用。数学学科分类:小学34M60;二级34E20, 34M35, 35A27, 35A30。作者的研究得到了日本科学促进会资助项目(20340028、21740098和21340029)的部分支持。102 Kamimoto, Kawai, Koike, and Takei方程，其势Q的形式为(0.2)，即具有大参数η的Whittaker方程，给出了具有一个简单拐点和一个简单极点的Schrödinger方程的wkb理论标准形式。在第2节中，我们注意到Whittaker方程中包含的参数α是一个无穷级数α(η) =∑k≥0 αkη−k (αk:一个常数)，当我们要强调α不是一个真正的常数而是一个无穷级数时，我们称这样的方程为∞-Whittaker方程。为了研究具有一个简单拐点和一个简单极势的Schrödinger方程的半全局问题，我们让简单极奇点与拐点合并，观察会出现什么样的方程。例如，如果我们让α在(0.2)中趋于零而γ保持不变会怎么样?有趣的是，结果方程就是我们所说的鬼方程(参见[Ko2]);我们一直在想，在整个WKB分析中，我们应该把这类鬼方程放在哪里。根据鬼方程的定义，它没有拐点(参见第1节注释1.1);尽管如此，鬼方程的WKB解仍然显示出与WKB解在拐点附近通常具有的奇点相似的奇点。奇点是由于势Q中η−k (k≥1)的系数中包含的奇点(详见[Ko2];有一个鬼(点)暂称“新”转折点。鉴于上述观察，我们把一个具有一个简单拐点和一个简单极点的Schrödinger方程看作是用一个简单极点项aq(x,a)/x扰动鬼方程得到的方程，其中a是一个复参数，q(x,a)是定义在(x,a) =(0,0)的邻域上的全纯函数。用这种方法得到的方程称为简单极点和简单拐点合并对方程，或简称为MPPT方程。准确地讲,我们称之为薛定谔方程(0.1)翻译一个MPPT方程如果它潜在的问还取决于一个辅助参数和的形式(0.3)Q = Q0 (x) x +η−1 Q1 (x) x +η−2 x2 Q2 (x),在Qj (x) (j = 0, 1, 2)附近的全纯(x) =(0, 0)和(x)满足以下条件(0.4)和(0.5):Q0 (0) = 0 = 0, (0.4) Q0 (x, 0) = c (0) 0 x + O (x)与c从0 0(0.5)一个常数不同。显然我们在a = 0处找到了一个鬼方程;更进一步，隐函数定理和假设(0.5)保证了满足(0.6)Q0 (x(a)， a) = 0的唯一全纯函数x(a)的存在性。假设(0.4)需要(0.7)x(a) = 0，如果a = 0，在MPPT算子103的wkb理论结构上，假设(0.5)保证，对于足够小的a(= 0)， x = x(a)是所讨论算子的一个简单拐点。正如术语“MPPT方程”所表明的那样，它在我们的上下文中是MTP方程的对应物。一个MTP方程，即[AKT4]中引入的合并拐点方程，根据定义包含两个简单拐点，当参数t趋于零时合并为一个双拐点;然而，在MPPT方程中，一个简单的极点和一个简单的转折点合并成一个鬼点，在η部分的最高阶(即零阶)中既没有观察到零，也没有观察到奇点。这两个概念的相似并不是表面上的。在第1节和第2节中实现了将MPPT方程简化为规范方程的方法，这种方法与将MTP方程简化为规范方程的方法类似。首先,在第一节我们构造一个翻译WKB-theoretic转换带来一个MPPT方程与参数为零到一个特定∞惠塔克方程,也就是说,惠特克∞方程与最高学位的一部分参数α(η)为零(即α(η)=∑k≥1αkη−k),然后在第二节我们构造转换的通用翻译(例如,= 0)MPPT方程的∞惠塔克方程在微扰级数的形式,从第1节中构造的转换开始。在第1节和第2节中，我们将注意力集中在问题的形式方面，并且在附录A和B中分别给出了出现在几个形式级数中的系数的增长顺序的估计。附录B中给出的估计的一个重要含义是它们通过Borel变换赋予形式变换作为微微分算子的解析意义。此外，如定理1.7和2.7所示，由此产生的微微分算子对多值解析函数(如borell变换的WKB解)的作用用特定类型的积分微分算子来描述;它的核函数包含一个无穷阶的x变量微分算子。因此，它在x变量中具有局部特征，而它适合于与y变量中的复苏现象相关的全局研究(例如，见[SKK]， [K]中的无限阶微分算子的概念;参见[AKT4]，它在精确WKB分析中首次使用了无限阶微分算子)。由于积分微分算子的定义域可以选择相对于参数a是一致的(见注2.3)，我们在第2节中的结果具有半全局特征，如注4.1所述。这种一致性是引入MPPT操作符概念时最重要的优点之一。值得强调的是，通过Borel变换，均匀性变得清晰可见。为了利用第2节的结果详细研究MPPT方程borel -变换WKB解的结构，我们首先在第3节中研究了Whittaker方程borel -变换WKB解的解析性质，然后在第4节中利用第3节的结果分析∞-Whittaker方程的borel -变换WKB解。第3节研究的基础是Koike [KoT]的最新结果，第4节的分析主要利用了系数{αk(a)}k≥0 104 Kamimoto, Kawai, Koike和Takei的参数α(a， η) =∑k≥0 αk(a)η−k的估计(B.3);∞-Whittaker方程中出现的这个无穷级数的效应被理解为作用于Whittaker方程borell变换的WKB解的微微分算子。结合第2节和第4节的所有结果，我们在第5节中总结了a = 0的MPPT方程borel变换WKB解的基本性质。1. 本节的目的是展示如何构造borel

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引用次数: 20

Connes-amenability of multiplier Banach algebras 乘子Banach代数的cones -amenability

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2010-01-01 DOI: 10.1215/0023608X-2009-003

B. Hayati, M. Amini

Let B be a Banach algebra with bounded approximate identity, and let M(B) be its multiplier algebra. If there exists a continuous linear injection B∗ → M(B) such that, for every b ∈ B and every u, v ∈ B∗, 〈u, vb〉B = 〈v, bu〉B , then M(B) is a dual Banach algebra and the following are equivalent: (i) B is amenable; (ii) M(B) is Connes amenable; (iii) M(B) has a normal, virtual diagonal.

设B是一个有界近似恒等式的巴拿赫代数，M(B)是它的乘数代数。如果存在一个连续线性注入B∗→M(B)，使得对于每个B∈B和每个u, v∈B∗，< u, vb > B = < v，但> B，则M(B)是对偶Banach代数，并且下列是等价的:(ii) M(B)是否符合Connes的规定;(iii) M(B)有一条法向虚对角线。

引用次数: 6

Relation between differential polynomials and small functions 微分多项式与小函数的关系

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2010-01-01 DOI: 10.1215/0023608X-2009-019

B. Belaïdi, A. Farissi

In this article, we discuss the growth of solutions of the second-order nonhomogeneous linear diﬀerential equation where a, b are complex constants and A j ( z ) (cid:2)≡ 0 ( j = 0 , 1) , and F (cid:2)≡ 0 are entire functions such that max { ρ ( A j ) ( j = 0 , 1) ,ρ ( F ) } < 1 . We also investigate the relationship between small functions and diﬀerential polynomials g f ( z ) = d 2 f (cid:2)(cid:2) + d 1 f (cid:2) + d 0 f , where d 0 ( z ) ,d 1 ( z ) ,d 2 ( z ) are entire functions that are not all equal to zero with ρ ( d j ) < 1 ( j = 0 , 1 , 2) generated by solutions of the above equation.

在本文中，我们讨论了二阶非齐次线性微分方程解的增长，其中a, b是复常数，且a j (z) (cid:2)≡0 (j = 0,1)， F (cid:2)≡0是使得max {ρ (a j) (j = 0,1)，ρ (F)} < 1的整函数。我们也调查之间的关系小函数和微分多项式g f d (z) = 2 f (cid: 2) (cid: 2) + d 1 f f (cid: 2) + d 0, 0 d (z), 2 d (z), d (z)是整个函数并非都是等于零,ρ(d j) < 1 (j = 0, 1, 2)由上述方程的解决方案。

引用次数: 7

The integral cohomology ring of $E_7/T$ $E_7/T$的整上同环

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2009-11-25 DOI: 10.3792/pjaa.86.64

Masaki Nakagawa

We give a complete description of the integral cohomology ring of the flag manifold E 8 /T, where E 8 denotes the compact exceptional Lie group of rank 8 and T its maximal torus, by the method due to Borel and Toda. This completes the computation of the integral cohomology rings of the flag manifolds for all compact connected simple Lie groups.

利用Borel和Toda的方法，给出了旗流形e8 /T的整上同环的完整描述，其中e8为秩8的紧化例外李群，T为其最大环面。这就完成了所有紧连通单李群的标志流形的整上同环的计算。

引用次数: 37

On Samelson products in Sp(n) 论Sp(n)中的Samelson产品

Q2 Mathematics

Journal of Mathematics of Kyoto University

Pub Date : 2009-09-24 DOI: 10.1215/KJM/1248983038

Tomoaki Nagao

引用次数: 2

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