Representations of Extended Affine Lie Algebras Coordinatized by Certain Quantum Tori

An irreducible representation of the extended affine Lie algebra of type A _n-1 coordinatized by a quantum torus of variables is constructed by using the Fock space for the principal vertex operator realization of the affine Lie algebra .     

Bosonic and fermionic representations of Lie algebra central extensions

Given any representation of an arbitrary Lie algebra g over a field K of characteristic 0, we construct representations of a central extension of g on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in H²(g;K). We illustrate these techniques with several concrete examples.     

Affine geometric crystals and limit of perfect crystals

For every non-exceptional affine Lie algebra, we explicitly construct a positive geometric crystal associated with a fundamental representation. We also show that its ultra-discretization is isomorphic to the limit of certain perfect crystals of the Langlands dual affine Lie algebra.

     

Hilbert Module Realization of the Square of White Noise and Finite Difference Algebras

We develop an approach to the representations theory of the algebra of the square of white noise based on the construction of Hilbert modules. We find the unique Fock representation and show that the representation space is the usual symmetric Fock space. Although we started with one degree of freedom we end up with countably many degrees of freedom. Surprisingly, our representation turns out to have a close relation to Feinsilver's finite difference algebra. In fact, there exists a holomorphic image of the finite difference algebra in the algebra of square of white noise. Our representation restricted to this image is the Boukas representation on the finite difference Fock space. Thus we extend the Boukas representation to a bigger algebra, which is generated by creators, annihilators, and number operators.     

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.     

The Fermion Model of Representations of Affine Krichever–Novikov Algebras

To a generic holomorphic vector bundle on an algebraic curve and an irreducible finite-dimensional representation of a semisimple Lie algebra, we assign a representation of the corresponding affine Krichever–Novikov algebra in the space of semi-infinite exterior forms. It is shown that equivalent pairs of data give rise to equivalent representations and vice versa.     

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.     

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.     

有限维非退化可解李代数的顶点算子代数

构造相应于非退化可解李代数g的顶点算子代数分两步进行,首先构造顶点代数.本文是在已经得到的相应于非退化可解李代数g的顶点代数(Vg(l,0),Y(V,1)上构造顶点算子代数.定义了非退化可解李代数g的Casimir算子Ω,给出了在伴随表示下Ω作用在g上是0及相关性质,并应用Ω定义出Vg(l,0)中元素ω,证明了Vg(l,0)关于ω的顶点算子YV(ω,x)的系数构成一个Virasoro代数-模,还证明了ω满足顶点算子代数定义中Virasoro-向量的所有公理.从而证得(Vg(l,0),Yv,1,ω)是一个顶点算子代数.     

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.     

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     

Quantum string theory in the space of states in an indefinite metric

The quantum string theory in the indefinite (pseudo-Hilbert) Fock representation is considered. It is shown that all anomalies in the four-dimensional theory of an open bosonic string are canceled for a particular choice of the vacuum state. Then the algebra of the reparametrization symmetry (in the covariant approach) and the algebra of rotations (in the light-cone gauge) become closed. The indefiniteness of the space of states leads to some nonphysical properties of the theory. Modifications of the model under consideration, which are necessary to construct a more realistic theory, are discussed.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 213–228, May, 1996.Translated by L. O. Chekhov.     

Wakimoto modules for twisted multi-loop algebra of type A1 × A1

We construct a class of modules for the twisted multi-loop algebra of type A1×A1 by applying Wakimoto free bosonic realization. We also discuss the structures and the irreducibility of the Fock space.     

Unidirectional Littelmann Paths for Crystals of Type A and Rank 2

Algebras and Representation Theory - For the Kashiwara crystal of a highest weight representation of an affine Lie algebra of type A and rank e, with highest weight Λ, there is a labeling by...     

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.     

CHARACTERISTIC CLASSES OF FLAGS OF FOLIATIONS AND LIE ALGEBRA COHOMOLOGY

We prove the conjecture by Feigin, Fuchs, and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given ag at the origin. The latter encodes characteristic classes of ags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan.Feigin, Fuchs, and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory. By the application of degeneration theorems of appropriate Hochschild-Serre spectral sequences, we avoid the need to use the methods of FFG, and moreover, we are able to describe all the symmetric powers at once.     

A unified view of some vertex operator constructions

We present a general vertex operator construction based on the Fock space for affine Lie algebras of typeA. This construction allows us to give a unified treatment for both the homogeneous and principle realizations of the affine Lie algebras as well as for some extended affine Lie algebras coordinatized by certain quantum tori.     

Wakimoto modules, opers and the center at the critical level

Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In this paper, we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc, giving another proof of the theorem of B. Feigin and the author.     

Tensor factorization and Spin construction for Kac-Moody algebras

In this paper we discuss the “Factorization phenomenon” which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into a tensor product of smaller representations of the subalgebra. We analyze this phenomenon for symmetrizable Kac-Moody algebras (including finite-dimensional, semi-simple Lie algebras). We present a few factorization results for a general embedding of a symmetrizable Kac-Moody algebra into another and provide an algebraic explanation for such a phenomenon using Spin construction. We also give some application of these results for semi-simple, finite-dimensional Lie algebras.We extend the notion of Spin functor from finite-dimensional to symmetrizable Kac-Moody algebras, which requires a very delicate treatment. We introduce a certain category of orthogonal g-representations for which, surprisingly, the Spin functor gives a g-representation in Bernstein-Gelfand-Gelfand category O. Also, for an integrable representation, Spin produces an integrable representation. We give the formula for the character of Spin representation for the above category and work out the factorization results for an embedding of a finite-dimensional, semi-simple Lie algebra into its untwisted affine Lie algebra. Finally, we discuss the classification of those representations for which Spin is irreducible.