Direct Unions of Lie Tori (Realization of Locally Extended Affine Lie Algebras)

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A₁, B, C, and BC.     

Lie series method for vector fields and Hamiltonian perturbation theory

We consider a rigorous Hamiltonian perturbation theory based on the transformation of the vector field of the system, realized by the Lie method. Such a perturbative technique presents some advantages over the standard one, which uses the transformation of the Hamilton functions. Indeed, the present method is simple, and furnishes quite detailed informations on the normal form. Moreover, it leads to estimates which are better and/or simpler than those of the scalar Lie methods. The perturbation method is presented with reference to two model problems, both pertaining to the realm of the well known Nekhoroshev theorem: the confining of actions for exponentially long times in a system of coupled harmonic oscillators, and an application to the so called problem of the realization of a holonomic constraint in classical mechanics.     

素环中广义导子的复合(英文)

Let R be a prime ring with a non-central Lie ideal L. In the present paper we show that if the composition DG of two generalized derivations D and G is zero on L, then DG must be zero on R. And we get all the possibilities for the composition of a couple of generalized derivations to be zero on a non-central Lie ideal of a prime ring.     

李三系的Frattini子系

李三系是从黎曼对称空间产生的三元运算的代数系统,近年来备受数学家们的重视.针对李三系的Frattini子系和基本李三系的问题进行了研究,给出了Frattini子系和基本李三系的一些性质,并证明了李三系的非嵌入定理,同时得到了幂零李三系是基本李三系的一个充要条件.     

On the dimensions of commutative subalgebras and subgroups

We consider the functions that bound the dimensions of finite-dimensional associative or Lie algebras in terms of the dimensions of their commutative subalgebras. It is proved that these functions have quadratic growth. As a result, we get similar estimates for the dimension of a Lie group with bounded dimensions of its Abelian Lie subgroups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 143–157, 2006.     

Global behavior of the heat kernel associated with certain sub-Laplacians on semisimple Lie groups

We prove global sharp estimates for the heat kernel related to certain sub-Laplacians on a real semisimple Lie group, from which we deduce an estimate for the corresponding Green function.     

Sous-laplaciens et densités centrées sur les groupes de Lie à croissance polynomiale du volume

We prove a Harnack inequality on connected Lie groups of polynomial volume growth. We use this inequality to study the large time behavior of the heat kernels associated to centered sub-Laplacians. Thus, we obtain Gaussian estimates and estimates of the type Berry—Esseen. We also obtain similar results for the convolution powers of centered densities.     

Cartan matrices for restricted Lie algebras

Consider a finite dimensional restricted Lie algebra over a field of prime characteristic. Each linear form on this Lie algebra defines a finite dimensional quotient of its universal enveloping algebra, called a reduced enveloping algebra. This leads to a Cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra. In this paper we show how to compare such Cartan matrices belonging to distinct linear forms. As an application we rederive and generalise the reciprocity formula first discovered by Humphreys for Lie algebras of reductive groups. For simple Lie algebras of Cartan type we see, for example, that the Cartan matrices for linear forms of non-positive height are submatrices of the Cartan matrix for the zero linear form.     

Symmetry Approaches for Reductions of PDEs, Differential Constraints and Lagrange-Charpit Method

Many methods for reducing and simplifying differential equations are known. They provide various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations between them have been noticed and in this note a unifying approach will be discussed. It is rather close to the differential constraint method, but we make this rigorous basing on recent advances in compatibility theory of non-linear overdetermined systems and homological methods for PDEs.      

Lower bounds for algebraic algorithms for nilpotent and solvable lie algebras

We obtain the lower bounds for the tensor rank for the class of nilpotent and solvable Lie algebras (in terms of dimensions of certain quotient algebras). These estimates, in turn, give lower bounds for the complexity of algebraic algorithms for this class of algebras. We adduce examples of attainable estimates for nilpotent Lie algebras of various dimensions.     

Norms of characters and Fourier series on compact Lie groups

We prove new estimates for the p norms of irreducible characters of compact Lie groups. These estimates are applied to give negative results on p mean convergence of Fourier series on compact Lie groups.     

具有小周期孔隙复合材料弹性结构的双尺度有限元分析

对于具有小周期孔隙复合材料弹性结构,在双尺度渐近分析理论结果的基础上提出了双尺度有限元计算格式,并给出了严格的误差估计.     

Relations among Lie-series transformations and isomorphisms between free Lie algebras

We study the subgroup generated by the exponentials of formal Lie series. We show three different ways to represent elements of this subgroup. These elements induce Lie-series transformations. Relations among these family of transformations furnish algorithms of composition. Starting from the Lazard elimination theorem and the Witt's formula, we show isomorphisms between some submodules of free Lie algebras. Combining different results, we also show that the homogeneous terms of the Hausdorff series H(a,b) freely generate the free Lie algebra L(a,b) without a line.     

Multiplicative nonholonomic/Newton-like algorithm

We construct new algorithms, which use the fourth order cumulant of stochastic variables for the cost function. The multiplicative updating rule here constructed is natural from the homogeneous nature of the Lie group and has numerous merits for the rigorous treatment of the dynamics. As one consequence, the second order convergence is shown. For the cost function, functions invariant under the componentwise scaling are chosen. By identifying points which can be transformed to each other by the scaling, we assume that the dynamics is in a coset space. In our method, a point can move toward any direction in this coset. Thus, no prewhitening is required.     

Estimations du Noyau de la Chaleur sur les Espaces Homogènes

We give upper and lower gaussian estimates for the heat kernel for canonical sub-laplacians on homogeneous spaces of Lie groups. We prove that homogeneous spaces of Lie groups of polynomial growth have the doubling property. We deduce Poincaré inequalities on these spaces.

     

Symbols and orbits for 3-step nilpotent Lie groups

Sufficient conditions are derived for L²-boundedness of a convolution operator on a 3-step nilpotent Lie group. This is achieved by producing estimates on the Kirillov symbol of the operator, and is closely linked to the co-adjoint orbit structure of the group. A structure theorem for 3-step nilpotent Lie groups with 1-dimensional center is proved.     

Group actions on chains of Banach manifolds and applications to fluid dynamics

This paper presents the theory of non-smooth Lie group actions on chains of Banach manifolds. The rigorous functional analytic spaces are given to deal with quotients of such actions. A hydrodynamical example is studied in detail.      

Generalized Burnside rings and group cohomology

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.     

李三系的中心扩张

李三系是从黎曼对称空间产生的三元运算的代数系统,近年来备受数学家们的重视.对李三系的中心扩张问题进行了研究,提出了Heisenberg李三系的概念,并对任意线性空间给出了构造Heisenberg李三系的一种方法.