Irreducible representations over the diamond Lie algebra

In this paper, we use Block’s results to classify irreducible modules over the diamond Lie algebra 𝔇. As a corollary, we also give a classification of irreducible modules over the Euclidean algebra 𝔢(2).     

From Lie algebra crossed modules to tensor hierarchies

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.     

一类李代数的结构

本文给出了复数域Ｃ上具有有限多个理想的有限维非可解李代数的结构为Ｌ＝Ｃｒ○＋Ｎ○＋Ｓ，其中Ｓ是Ｌ的Ｌｅｖｉ因子，Ｎ是Ｌ的幂零根基，Ｃｒ○＋Ｎ是Ｌ的根基，ａｄｒ是Ｎ的半单纯线性变换，［ｒ，Ｓ］＝０；还给出了这类李代数的一些重要性质．     

Projective Limits of Finite-Dimensional Lie Groups

For a topological group G we define N to be the set of all normalsubgroups modulo which G is a finite-dimensional Lie group.Call G a pro-Lie group if, firstly, G is complete, secondly,N is a filter basis, and thirdly, every identity neighborhoodof G contains some member of N. It is easy to see that everypro-Lie group G is a projective limit of the projective systemof all quotients of G modulo subgroups from N. The converseimplication emerges as a difficult proposition, but it is shownhere that any projective limit of finite-dimensional Lie groupsis a pro-Lie group. It is also shown that a closed subgroupof a pro-Lie group is a pro-Lie group, and that for any closednormal subgroup N of a pro-Lie group G, for any one parametersubgroup Y : R G/N there is a one parameter subgroup X : R G such that X(t) N = Y(t) for any real number t. The categoryof all pro-Lie groups and continuous group homomorphisms betweenthem is closed under the formation of all limits in the categoryof topological groups and the Lie algebra functor on the categoryof pro-Lie groups preserves all limits and quotients. 2000 MathematicsSubject Classification 22E65, 22D05, 22E20, 22A05, 54B35.     

Class of representations of skew derivation Lie algebra over quantum torus

Let L be the skew derivation Lie algebra of the quantum torus ℂ_q. In this paper, we give a class of irreducible representations for L with infinite dimensional weight spaces.      

Deformations of the infinite-dimensional Lie algebra L 3

For the nilpotent infinite-dimensional Lie algebra L ₃, we compute the second cohomology group H ²(L ₃, L ₃) with coefficients in the adjoint module. Nontrivial cocycles are found in closed form, and Massey powers are computed for them.     

Automorphisms Group of an Invariant Subalgebra of a Reductive Lie Algebra

We prove that any automorphism of an invariant subalgebra of a reductive Lie algebra over a field of zero characteristic is a standard automorphism. Mathematics Subject Classifications (2000) 17B20, 17B40.     

Whittaker modules for the derivation Lie algebra of torus with two variables

Let L be the derivation Lie algebra of C[t±11,t±12].Given a triangle decomposition L=L~+⊕h⊕L~-,we define a nonsingular Lie algebra homomorphism ψ:L~+→C and the universal Whittaker L-module W_ψ of type ψ.We obtain all Whittaker vectors and submodules of W_ψ.Moreover,all simple Whittaker L-modules of type ψ are determined.     

Krull dimension of the enveloping algebra of a semisimple Lie algebra

Let be a complex semisimple Lie algebra and be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of is equal to the dimension of a Borel subalgebra of .

     

Special polynomials in free framed Lie algebra

A framed Lie algebra is an algebra with two operations which is a Lie algebra with respect to one of these operations. A basic example is a Lie algebra of vector fields on a manifold with connection where the covariant derivative serves as an additional operation. In a free framed Lie algebra, we distinguish a set of special polynomials that geometrically correspond to invariantly defined tensors. A necessary condition of being special is derived, and we presume that this condition is also sufficient. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 571–583, September–October, 2008.     

The centralizer of an element in a Lie algebra of type L

In this paper, we investigate the Lie algebra L(A, α, δ) of type L and obtain the respective sufficient conditions for L(A, α,δ) to be semisimple, and for Z(ω) = Fω as well, where 0 ≠ ω ∈ L(A, α, δ) and Z(ω) is the centralizer of ω.     

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.     

Restricted Lie 2-algebras

In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A(m)-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.