Solvable Lie Algebras, Lie Groups andPolynomial Structures

In this paper, we study polynomial structures by starting on the Lie algebra level, thenpassing to Lie groups to finally arrive at the polycyclic-by-finite group level. To be more precise,we first show how a general solvable Lie algebra can be decomposed into a sum of two nilpotentsubalgebras. Using this result, we construct, for any simply connected, connected solvable Lie groupG of dim n, a simply transitive action on R ⁿ which is polynomial and of degree n³. Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group , which is of degree h()³ on almost the entire group (h () being the Hirsch length of ).     

Remarks on Gelfand Pairs Associated to Non-Type-I Solvable Lie Groups

LetSbe a connected and simply connected unimodular solvable Lie group andKa connected compact Lie group acting onSas automorphisms. We call the pair (K S) a Gelfand pair if the Banach ∗-algebraL¹_K(S) of allK-invariant integrable functions onSis a commutative algebra. In this paper we give a necessary and sufficient condition for the pair (K; S) to be a Gelfand pair using the representation theory of non-type-I solvable Lie groups. For a Gelfand pair (K; S) we realize all irreducibleK-spherical representations ofK?Sfrom irreducible unitary representations ofS.     

Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.     

Solitary Solvable Groups

A subgroup H of a group G is a solitary subgroup of G if G does not contain another isomorphic copy of H. Combining together the concepts of solitary subgroups and solvable groups, we define (normal) solitary solvable groups and (normal) strongly solitary solvable groups. We derive several results that hold for these groups and we discuss classes of groups that, under certain hypotheses, are (normal) solitary solvable and (normal) strongly solitary solvable. We also derive several results about p-groups that are solitary solvable.     

The hypoelliptic Laplacian on a compact Lie group

Let G be a compact Lie group, and let g be its Lie algebra. In this paper, we produce a hypoelliptic Laplacian on G×g, which interpolates between the classical Laplacian of G and the geodesic flow. This deformation is obtained by producing a suitable deformation of the Dirac operator of Kostant. We show that various Poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory. The paper was motivated by papers by Atiyah and Frenkel, in connection with localization formulas in equivariant cohomology and with Kac's character formulas for affine Lie algebras. In a companion paper, we will use similar methods in the context of Selberg's trace formula.     

关于可解李三超系

给出了关于李三超系的一些基本概念和性质,包括:(1)可解李三超系的任意包络李超代数都是可解的;(2)如果一个李蔓超系有可解的包络李超代数,则它也是可解的;(3)一个李三超系T是幂零的当且仅当它的标准嵌入李超代数L(T)是幂零的;(4)L(T)的中心等于T的中心;(5)李三超系上的右不变超对称双线性型可唯一地扩张到它的标准嵌入李超代数上.     

RESEARCH ANNOUNCEMENTS——Structure of Solvable Quadratic Lie Algebras

Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics^[10，12，13]. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras. In this paper, we study solvable quadratic Lie algebras. In Section 1, we study quadratic solvable Lie algebras whose Cartan subalgebras consist of semi-simple elements. In Section 2,we present a procedure to construct a class of quadratic Lie algebras, and we can exhaust all solvable quadratic Lie algebras in such a way. All Lie algebras mentioned in this paper are finite dimensional Lie algebras over a field F of characteristic 0.     

Intermediate Growth of Solvable Lie Superalgebras

Finitely generated solvable Lie algebras have an intermediate growth between polynomial and exponential. Recently the second author suggested the scale to measure such an intermediate growth of Lie algebras. The growth was specified for solvable Lie algebras F(A ^q , k) with a finite number of generators k, and which are free with respect to a fixed solubility length q. Later, an application of generating functions allowed us to obtain more precise asymptotic. These results were obtained in the generality of polynilpotent Lie algebras. Now we consider the case of Lie superalgebras; we announce that main results and describe the methods. Our goal is to compute the growth for F(A ^q , m, k), the free solvable Lie superalgebra of length q with m even and k odd generators. The proof is based upon a precise formula of the generating function for this algebra obtained earlier. The result is obtained in the generality of free polynilpotent Lie superalgebras. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Structure of Solvable Rational Groups

R. Gow proved that the order of a solvable rational group isdivisible only by the primes 2, 3 and 5. In this paper it isproved that in a solvable rational group the Sylow 5-subgroupis always normal and elementary Abelian. Moreover, the structureof rational {2, 5}-groups is described in detail. 2000 MathematicsSubject Classification 20C15, 20C20, 20E34, 20E45.     

可解群的大轨道

G可解群,G忠实的作用在有限群H上,且(|G|,|H|)=1,那么存在x,y∈H满足C_G(x)∩C_G(y)=1,即G有大轨道.     

关于可解群p—块的几个结果

主要结果是如下定理:设G是有限可解群使得G/F(G)是奇阶A-群,又设p是一个素数且G不含截断q~(pn):(Z_m:Z_p)。其中q~(pn):(Z_m:Z_n))是初等交换q-群q~(pn)被Z_m:Z_p的扩张,而m=(q~(pn)-1)/(q~n-1)。则G有亏数零p-块的充要条件是O_p(G)=1。     

Factoring a Lie Group into a Compactly Embeddedand a Solvable Subgroup

 Let G be a real connected Lie group. A subgroup K is called compactly embedded if the closure of Ad(K) is compact in Aut(). If K is, in addition, maximal with respect to this property, then there exists a solvable subgroup S containing the nilradical such that and is the one-component of the center of G. (Received 1 June 1999; in revised form 28 December 1999)     

Two Kinds of Solvable Complete Lie Algebras

In this paper,we will give the definition of completable nilpotent Lie algebras,discuss its decomposition and prove that the heisenberg algebras and extensions of abelian quadratic Lie algebras are all completable nilpotent Lie algebras.