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.     

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.     

Simple left-symmetric algebras with solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie groupG correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we studysimple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied.     

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.     

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.     

The degeneracy of extended affine Lie algebras

We construct degenerate extended affine Lie algebras from a given nondegenerate extended affine Lie algebra and show that all degenerate extended affine Lie algebras are obtained in this way. Received: 21 January 1997     

Simple left-symmetric algebras with¶solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group {G} correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we study simple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied. Received: 10 June 1997 / Revised version: 29 September 1997     

Holomorphic affine vector fields on weil bundles

We obtain necessary and sufficient conditions for a holomorphic vector field to be affine for a holomorphic linear connection defined on aWeil bundle. We also prove that the Lie algebra over R of holomorphic affine vector fields for the natural prolongation of a linear connection from the base to theWeil bundle is isomorphic to the tensor product of theWeil algebra by the Lie algebra of affine vector fields on the base.     

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(?).     

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．     

The structure of integrable Lax flows on nonlocal manifolds: Dynamic systems with sources

We consider a dynamic system on the extended phase space to the initial Lie algebra and study its generalized Hamiltonian and integrability in the cases when the initial Lie algebra coincides with the Grassmann algebra of pseudodifferential operators on the real line and on the centrally extended affine Lie algebra.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 106–115.     

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.     

Polynomial Representations and Categorifications of Fock Space

The rings of symmetric polynomials form an inverse system whose limit, the ring of symmetric functions, is the model for the bosonic Fock space representation of the affine Lie algebra. We show that this limit naturally carries an action of the affine Lie algebra (in the sense of Rouquier), thereby obtaining a family of categorifications of the bosonic Fock space representation.     

Commutative post-Lie algebra structures on Kac–Moody algebras

We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore, we show that all commutative post-Lie algebra structures on affine Kac–Moody Lie algebras are “almost trivial”.     

Quiver varieties and Frenkel–Kac construction

An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type A the former is obtained by Frenkel–Kac construction from the latter.     

Representations of Loop Kac-Moody Lie Algebras

We study representations of the Loop Kac-Moody Lie algebra 𝔤 ?A, where 𝔤 is any Kac-Moody algebra and A is a ring of Laurent polynomials in n commuting variables. In particular, we study representations with finite dimensional weight spaces and their graded versions. When we specialize 𝔤 to be a finite dimensional or affine Lie algebra we obtain modules for toroidal Lie algebras.     

Left invariant special Kähler structures

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial 1-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.     

Affine varieties and lie algebras of vector fields

In this article, we associate to affine algebraic or local analytic varieties their tangent algebra. This is the Lie algebra of all vector fields on the ambient space which are tangent to the variety. Properties of the relation between varieties and tangent algebras are studied. Being the tangent algebra of some variety is shown to be equivalent to a purely Lie algebra theoretic property of subalgebras of the Lie algebra of all vector fields on the ambient space. This allows to prove that the isomorphism type of the variety is determinde by its tangent algebra.