The Core of a Locally Extended Affine Lie Algebra

Locally extended affine Lie algebras are a general version of extended affine Lie algebras. In this article, we completely describe the structure of the core of a locally extended affine Lie algebra. We prove that the core of a locally extended affine Lie algebra is a direct limit of Lie tori.     

A Generalization of Extended Affine Lie Algebras

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.     

Extended affine Lie superalgebras and their affinizations

Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process.     

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.     

Infinite Rank Affine Lie Algebras

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D_4型广义扭仿射李代数及其模

本文是[3]的继续,将讨论D4型的广义扭仿射李代数及其表示理论;证明作用在其不可约模上的一类算子的局部幂零性.     

无限秩仿射李代数的cartan子代数的共轭性

本文给出了无限秩仿射李代数的某种类型的Cartan子代数的定义,并证明了这种Cartan子代数在无限秩仿射李代数的某种类型的自同构下的共轭性.     

Affine Poisson structures

We establish a one-to-one correspondence between the set of all equivalence classes of affine Poisson structures (defined on the dual of a finite dimensional Lie algebra) and the set of all equivalence classes of central extensions of the Lie algebra by ℝ. We characterize all the affine Poisson structures defined on the duals of some lower dimensional Lie algebras. It is shown that under a certain condition every Poisson structure locally looks like an affine Poisson structure. As an application, we show the role played by affine Poisson structures in mechanics. Finally, we prove some involution theorems.     

Refinement of Ado's Theorem in Low Dimensions and Application in Affine Geometry

In this article, we construct a faithful representation with the lowest dimension for every complex Lie algebra in dimension ≤ 4. In particular, in our construction, in the case that the faithful representation has the same dimension of the Lie algebra, it can induce an étale affine representation with base zero which has a natural and simple form and gives a compatible left-symmetric algebra on the Lie algebra. Such affine representations do not contain any nontrivial one-parameter subgroups of translation.     

Toroidal李代数上的Poisson代数结构

非交换的Poisson代数同时具有结合代数和李代数两种代数结构,而结合代数和李代数之间满足所谓的Leibniz法则．文中确定了Toroidal李代数上所有的Poisson代数结构,推广了仿射Kac-Moody代数上相应的结论．     

Modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules,and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated,and several results are generalized from the affine Lie algebras. In particular,imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.     

Complete Reducibility for the Twisted Affine Lie Algebras

We prove complete reducibility theorem for integrable modules for the twisted affine Lie algebras where the central element acts non-trivially.     

广义仿射李代数上的非交换Poisson代数结构

非交换的Poisson代数同时具有(未必交换的)结合代数和李代数两种代数结构,且结合代数和李代数之间满足所谓的Leibniz法则.本文确定了一般广义仿射李代数上所有的Poisson代数结构.     

Derivation Algebras of Toric Varieties

We study the Lie algebra of derivations of the coordinate ring of affine toric varieties defined by simplicial affine semigroups and prove the following results:Such toric varieties are uniquely determined by their Lie algebra if they are supposed to be Cohen–Macaulay of dimension 2 or Gorenstein of dimension =1.In the Cohen–Macaulay case, every automorphism of the Lie algebra is induced from a unique automorphism of the variety.Every derivation of the Lie algebra is inner.     

Tits-Kantor-Koecher李代数的Wakimoto表示

每一个Jordan代数都对应了一个Tits-Kantor-Koecher李代数．在扩张仿射李代数的分类中[1],A_1型李代数的分类依赖于欧氏空间上半格给出的Tits-Kantor-Koecher李代数．另外在相似的意义下,二维欧氏空间R~2中只有两个半格．设S是R~2上的任一半格,T(S)为半格S对应的Jordan代数,G(T(S))为相应的Tits-Kantor-Koecher李代数．利用Wakimoto自由场的方法给出李代数G(T(S))的一类顶点表示．     

NON-COMMUTATIVE POISSON ALGEBRA STRUCTURES ON LIE ALGEBRA sln（Cq^-） WITH NULLITY M

Non-commutative Poisson algebras are the algebras having both an associa- tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln（Cq^-） are determined.     

Irreducible Representations for the Baby Tits–Kantor–Koecher Algebra

The baby Tits–Kantor–Koecher (TKK) algebra constructed from the smallest (nonlattice) semilattice is related to the “smallest” extended affine Lie algebras other than the finite dimensional simple Lie algebras and the affine Kac–Moody algebras. In this article, we classify the finite dimensional irreducible representations for the baby TKK algebra. It turns out that such representations can be lifted from modules of direct sums of finitely many copies of the simple Lie algebra sp ₄(?).     

着色李超代数与左着色对称结构

本文研究了着色李超代数上的左着色对称结构问题.利用着色李超代数的两种仿射表示和1-上同调群,得出左着色对称结构存在的几个充分或必要条件,推广了文[2]的结论.     

Differential equations uniquely determined by algebras of point symmetries

We continue to investigate strongly and weakly Lie remarkable equations, which we defined in a recent paper. We consider some relevant algebras of vector fields on ℝ^k (such as the isometric, affine, projective, or conformal algebras) and characterize strongly Lie remarkable equations admitted by the considered Lie algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 486–494, June, 2007.