简约李代数和局部李代数的生成元

证明了具有n-维中心的简约李代数的生成元的最小个数是n或n+1(n≥1),也证明了满足一定约束条件的局部李代数能由两个元素生成.     

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

On Complete Lie Algebras

In this article, the author gives two methods to construct complete Lie algebras. Both methods show that the derivation algebras of some Lie algebras are complete.     

Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Let \(\mathfrak g\) be a semisimple Lie algebra over a field \(\mathbb K\), \(\text{char}\left( \mathbb{K} \right)=0\), and \(\mathfrak g_1\) a subalgebra reductive in \(\mathfrak g\). Suppose that the restriction of the Killing form B of \(\mathfrak g\) to \(\mathfrak g_1 \times \mathfrak g_1\) is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra \(\mathfrak h_1\) of \(\mathfrak g_1\) there is a unique Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) containing \(\mathfrak h_1\); ( 2) \(\mathfrak g_1\) is self-normalizing in \(\mathfrak g\); ( 3) The B-orthogonal \(\mathfrak p\) of \(\mathfrak g_1\) in \(\mathfrak g\) is simple as a \(\mathfrak g_1\)-module for the adjoint representation. We give some answers to this natural question: For which pairs \((\mathfrak g,\mathfrak g_1)\) do ( 1), ( 2) or ( 3) hold? We also study how \(\mathfrak p\) in general decomposes as a \(\mathfrak g_1\)-module, and when \(\mathfrak g_1\) is a maximal subalgebra of \(\mathfrak g\). In particular suppose \((\mathfrak g,\sigma )\) is a pair with \(\mathfrak g\) as above and σ its automorphism of order m. Assume that \(\mathbb K\) contains a primitive m-th root of unity. Define \(\mathfrak g_1:=\mathfrak g^{\sigma}\), the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) \((\mathfrak g,\mathfrak g_1)\) satisfies ( 1); (b) For m prime, \((\mathfrak g,\mathfrak g_1)\) satisfies ( 2).     

关于完备李群与完备李代数

孟道骥等对完备李代数作了系统的研究并已获得很多基本和重要的结果。本文给出完备李群与完备李代数的某些关系。     

On Contractions of Lie Algebras

It is well-known that application of invariant functions in various fields of pure or applied mathematics, physics and computer calculations is certainly of interest. In this paper, we study application of the invariant functions of Lie algebras, named \(\psi \) and \(\varphi \) function, respectively, to contractions of some lower-dimensional Lie algebras, with the goal of contributing to progress in any of these fields, particularly in physics and engineering.     

Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

On Enveloping Lie Algebras of Hyporeductive Triple Algebras

The notion of a hypoderivation of binary-ternary algebras is introduced. A hypoderivation is a generalization both of a derivation and a pseudoderivation of such algebras. From the external direct sum of a hyporeductive triple algebra (h.t.a.) with the vector space of pairs constituted by hypoderivations and their companions, a Lie algebra with a hyporeductive decomposition (and accordingly a hyporeductive pair) enveloping the given h.t.a. is constructed. A nontrivial 3-dimensional Lie algebra with hyporeductive decomposition is presented. Examples of h.t.a. are also given.     

色李代数中的李定理

作者证明了无挠群上的色李代数的李定理, 同时, 也给出例子说明 无挠性是必要的     

A-扩张Lie Rinehart代数

The purpose of this paper is to give a brief introduction to the category of Lie Rinehart algebras and introduces the concept of smooth manifolds associated with a unitary, commutative,associative algebra A.It especially shows that the A-extended algebra as well as the action algebra can be realized as the space of A-left invariant vector fields on a Lie group,analogous to the well known relationship of Lie algebras and Lie groups.     

On Polynomial Representations of Lie Algebras

We study polynomial representations of finite dimensional (R or C) Lie algebras. As a total classification, we show that there are altogether three types of such nontrivial representations and give their subtle structures.     

On Lie Algebras with Wild Automorphisms

We exhibit a class of Lie algebras (which contains, in particular, free polynilpotent Lie algebras) that have wild automorphisms, i.e., automorphisms which are not induced by automorphisms of a free Lie algebra.     

On Auslander-Reiten Quivers of Enveloping Algebras of Restricted Lie Algebras

Let (L, [p]) be a finite dimensional restricted Lie algebra over an algebraically closed field F of characteristic p ≥ 3, X ∈ L* a linear form. In this article we study the Auslander-Reiten quivers of certain blocks of the reduced enveloping algebra u(L,x). In particular, it is shown that the enveloping algebras of supersolvable Lie algebras do not possess AR-components of Euclidean type.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

广义矩阵代数上的李导子

研究了广义矩阵代数上的一类李导子,证明了广义矩阵代数上李导子可以表示成一个导子和一个中心映射之和,并将这个结果应用到全矩阵代数上.     

半单李代数的单纯作用

本文证明了任何半单李代数(或者李群)在连通光滑流形上的非平凡单纯作用一定没有驻点.而且有效作用的那部分必定是同构于sl(2,R)(或者SL(2,R))的理想.