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.     

Characteristically nilpotent Lie algebras

Translated from Algebra i Logika, Vol. 28, No. 6, pp. 722–737, November–December, 1989.     

Invariant Lie Algebras and Lie Algebras with a Small Centroid

A subalgebra of a Lie algebra is said to be invariant if it is invariant under the action of some Cartan subalgebra of that algebra. A known theorem of Melville says that a nilpotent invariant subalgebra of a finite-dimensional semisimple complex Lie algebra has a small centroid. The notion of a Lie algebra with small centroid extends to a class of all finite-dimensional algebras. For finite-dimensional algebras of zero characteristic with semisimple derivations in a sufficiently broad class, their centroid is proved small. As a consequence, it turns out that every invariant subalgebra of a finite-dimensional reductive Lie algebra over an arbitrary definition field of zero characteristic has a small centroid.     

Polynomial Lie Algebras

We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings k[x ₁,...,x _n]/(f ₁,...,f _n) is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types A,B,C, D, and E ₆.     

A remark on left invariant metrics on compact Lie groups

We show that a left invariant metric on a compact Lie group G with Lie algebra has some negative sectional curvature if it is obtained by enlarging a biinvariant metric on a subalgebra , unless the semi-simple part of is an ideal of This answers a question raised in [8]. Received: 7 May 2007     

Lie Algops and Simple Lie Algebras

ABSTRACT

Pairs (A, L) with A a commutative algebra and L a Lie algebra acting on A by derivations, called Lie algops, are studied as algebraic structures over arbitrary fields of arbitrary characteristic. Lie algops possess modules and tensor products—and are considered with respect to a central simple theory.

The simplicity problem of determining the faithful unital simple Lie algops ( A, L ) is of interest since the corresponding Lie algebras AL are usually simple (Jordan, 2000 Jordan , D. A. ( 2000 ). On the simplicity of Lie algebras of derivations of commutative algebras . J. Algebra 228 : 580 – 585 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). For locally finite Lie algops, and up to purely inseparable descent, this problem reduces by way of closures to the closed central simplicity problem of determining those which are closed central simple.

The simplicity and representation theories for locally nilpotent separably triangulable unital Lie algops are of particular interest because they relate to the problems of classifying simple Lie algebras of Witt type and their representations. Of these, the simplicity theory reduces to that of Jordan Lie algops.

The main Theorems 7.3 and 7.4 reduce the simplicity and representation theories for Jordan Lie algops to the simplicity and representation theories for simple nil and toral Lie algops.     

Decompositions of nilpotent Lie algebras

It is proved that decompositions of nilpotent Lie algebras are global. In the complex case, nilpotency is also a necessary condition for every decomposition to be global. The results obtained are applied to the classification of complex homogeneous spaces of simply connected nilpotent Lie groups.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 27–30, January, 1978.In conclusion, the author would like to thank A. L. Onishchik for his interest in this research.     

Representations of nilpotent Lie algebras

Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic p 3 3, c ? L^*p \geq 3, \chi \in L^* a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra u(L,c)u(L,\chi ). It is shown that u(L,c)u(L,\chi ) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of u(L,c)u(L,\chi ) are of types \Bbb Z [A_￥ ]\Bbb Z [A_\infty ] or \Bbb Z [A_n]/(t)\Bbb Z [A_n]/(\tau ).