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1.
Let G be a connected Lie group, with Lie algebra . In 1977, Duflo constructed a homomorphism of -modules , which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group G. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara–Vergne conjecture for any Lie group G. (2) We give a reformulation of the Kashiwara–Vergne property in terms of Lie algebra cohomology. As a direct corollary, one obtains the algebra isomorphism , as well as a more general statement for distributions.  相似文献   

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We show that the Kashiwara–Vergne (KV) problem for quadratic Lie algebras (that is, Lie algebras admitting an invariant scalar product) reduces to the problem of representing the Campbell–Hausdorff series in the form ln(exey)=x+y+[x,a(x,y)]+[y,b(x,y)], where a(x,y) and b(x,y) are Lie series in x and y. This observation explains the existence of explicit rational solutions of the quadratic KV problem, whereas constructing an explicit rational solution of the full KV problem would probably require the knowledge of a rational Drinfeld associator. It also gives, in the case of quadratic Lie algebras, a direct proof of the Duflo theorem (implied by the KV problem). To cite this article: A. Alekseev, C. Torossian, C. R. Acad. Sci. Paris, Ser. I 347 (2009).  相似文献   

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We define a family KV(g,n+1) of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with n+1 boundary components. The problem KV(0,3) is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to KV(g,n+1) for arbitrary g and n. The key point is the solution to KV(1,1) based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra g(g,n+1). In more detail, we show that every solution to KV(g,n+1) induces a Lie bialgebra isomorphism between g(g,n+1) and its associated graded grg(g,n+1). For g=0, a similar result was obtained by G. Massuyeau using the Kontsevich integral. For g1, n=0, our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.  相似文献   

4.
The Kashiwara–Vergne (KV) conjecture states the existence of solutions of a pair of equations related with the Campbell–Baker–Hausdorff series. It was solved by Meinrenken and the first author over ℝ, and in a formal version, by two of the authors over a field of characteristic 0. In this paper, we give a simple and explicit formula for a map from the set of Drinfeld associators to the set of solutions of the formal KV equations. Both sets are torsors under the actions of prounipotent groups, and we show that this map is a morphism of torsors. When specialized to the KZ associator, our construction yields a solution over ℝ of the original KV conjecture.  相似文献   

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We consider the following Brezis–Nirenberg problem on S3
?ΔS3u=λu+u5inD,u>0inDandu=0on ?D,
where D is a geodesic ball on S3 with geodesic radius θ1, and ΔS3 is the Laplace–Beltrami operator on S3. We prove that for any λ<?34 and for every θ1<π with π?θ1 sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264–279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394]. To cite this article: W. Chen, J. Wei, C. R. Acad. Sci. Paris, Ser. I 341 (2005).  相似文献   

7.
In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.  相似文献   

8.
Following a previous result stating their equivalence under constant advection speed, Semi-Lagrangian and Lagrange–Galerkin schemes are compared in this paper in the situation of variable coefficient advection equations. Once known that Semi-Lagrangian schemes can be proved to be equivalent to area-weighted Lagrange–Galerkin schemes via a suitable definition of the basis functions, we will further prove that area-weighted Lagrange–Galerkin schemes represent a “small” (more precisely, an $O(\Delta t$ )) perturbation of exact Lagrange–Galerkin schemes. This equivalence implies a general result of stability for Semi-Lagrangian schemes.  相似文献   

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We characterize CR functions on planar domains and real hypersurfaces in C2 in terms of analytic extendibility into attached analytic discs. It is done by studying propagation, from the boundary into interior, of degeneracy of CR foliations of solid torus-like manifolds. In particular, we answer, for smooth functions, two open questions mentioned in the title: about characterization of analytic functions in the complex plane and about characterization of boundary values of holomorphic functions in bounded domains in Cn. To cite this article: M. Agranovsky, C. R. Acad. Sci. Paris, Ser. I 343 (2006).  相似文献   

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Given k1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2k-variable linear Diophantine equation
i=1k(xi?yi)=b
is (2k?1)-regular. This is best possible, since Fox and Kleitman showed that for all b1, this equation is not 2k-regular. While the conjecture has recently been settled for all k2, here we focus on the case k=3 and determine the degree of regularity of the corresponding equation for all b1. In particular, this independently confirms the conjecture for k=3. We also briefly discuss the case k=4.  相似文献   

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Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg\Bbbk=\mathfrakg\mathbbZ?\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbFp{\mathbb{F}}_{p}. Let G\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G\Bbbk)=\mathfrakg\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg\Bbbk)G\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B n , n≥3, D n , n≥4, E6, E7, E8 or F4. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B n , n≥3, D n , n≥4, E6, E7, E8 or F4, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.  相似文献   

19.
We introduce the notion of a generalized interval exchange induced by a measurable k-partition of [0,1). can be viewed as the corresponding restriction of a nondecreasing function on ℝ with . A is called λ-dense if λ(A i ∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that . We give necessary and sufficient conditions for this equality to hold. We show that for each integer m≥2, such that 3∤2m+1, there exist 2 and 3 non λ-dense partitions A and B of [0,1), corresponding to the interval exchanges on 2m intervals, for which and commute.  相似文献   

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