Finite-Dimensional Half-Integer Weight Modules over Queer Lie Superalgebras

We give a new interpretation of representation theory of the finite-dimensional half-integer weight modules over the queer Lie superalgebra \({\mathfrak{q}(n)}\). It is given in terms of the Brundan’s work on finite-dimensional integer weight \({\mathfrak{q}(n)}\)-modules by means of Lusztig’s canonical basis. Using this viewpoint we compute the characters of the finite-dimensional half-integer weight irreducible modules. For a large class of irreducible modules whose highest weights are of special types (i.e., totally connected or totally disconnected) we derive closed-form character formulas that are reminiscent of the Kac–Wakimoto character formula for basic Lie superalgebras.     

Integrable Highest Weight Modules over Affine Superalgebras and Appell's Function

We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level 1 case. The analysis of this construction shows, in particular, that in the simplest case of the sℓ (2|1) level 1 affine superalgebra the characters are expressed in terms of the Appell elliptic function. Our results demonstrate that the representation theory of affine Lie superalgebras is quite different from that of affine Lie algebras. Received: 17 April 2000 / Accepted: 7 July 2000     

Interwining Operators for Vertex Representations of Toroidal Lie Algebras

The construction of the vertex representation of the toroidal Lie algebra ^[n] depends on the way of labelling the points in the dual ⁿ of the torus Tⁿ. Thus there is a built-in symmetry of the vertex representation with respect to the symmetry of Zⁿ. In conjunction with this, the energy operator L₀ gives rise to intertwining operators which reflect the symmetry of the vertex representation with respect to S¹ action on .     

The O'RAIFEARTAIGH Theorem for the Generalized Lie Algebras

The generalization of O'RAIFEARTAIGH'S theorem about mass splitting to the case of an algebra containing both commutators and anticommutators is given.     

SAYD Modules over Lie-Hopf Algebras

In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.     

On Highest Weight Modules Over Elliptic Quantum Groups

The purpose of this Letter is to define and construct highest weight modules for Felder's elliptic quantum groups. This is done by using exchange matrices for intertwining operators between modules over quantum affine algebras.     

Reduction Groups and Automorphic Lie Algebras

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts.On leave from, L.D. Landau Institute for Theoretical Physics Chernogolovka, Russia     

Simple Modules over the Higher-Rank Super-Virasoro Algebras

It is proved that uniformly bounded simple modules over higher-rank super-Virasoro algebras are modules of the intermediate series, and that simple modules with finite dimensional weight spaces are either modules of the intermediate series or generalized highest weight modules.     

A Toroidal Magnetic Field Theorem

In the framework of magnetohydrodynamics the kinematic dynamo problem in a spherical fluid volume as well as in a plane layer is considered. On the premises of a purely toroidal magnetic field a nonlinear evolution equation for the toroidal scalar is derived. In this equation the flow field is constrained in such a way that no poloidal magnetic field can arise, but is otherwise arbitrary; the magnetic diffusivity is assumed to be spherically (horizontally, resp.) symmetric. Solutions of this problem are of particular interest since the magnetic field is confined to the fluid volume and therefore invisible to an external observer. It is proved in this paper that the maximum norm of smooth solutions of this equation decays exponentially fast to zero. Thus, dynamo solutions, i.e. nondecaying solutions, of this type do not exist.     

Classification of Harish-Chandra Modules over the Higher Rank Virasoro Algebras

We classify the Harish-Chandra modules over the higher rank Virasoro and super-Virasoro algebras: It is proved that a Harish-Chandra module, i.e., an irreducible weight module with finite weight multiplicities, over a higher rank Virasoro or super-Virasoro algebra is either a module of the intermediate series, or a finitely-dense module. As an application, it is also proved that an indecomposable weight module with finite weight multiplicities over a generalized Witt algebra is either a uniformly bounded module (i.e., a module with weight multiplicities uniformly bounded) with all nonzero weights having the same multiplicity, or a finitely-dense module, as long as the generalized Witt algebra satisfies one minor condition.This work is supported by a NSF grant 10171064 of China and two grants ``Excellent Young Teacher Program' and ``Trans-Century Training Programme Foundation for the Talents' from Ministry of Education of China.     

A Basis for Representations of Symplectic Lie Algebras

A basis for each finite-dimensional irreducible representation of the symplectic Lie algebra ¤(2n) is constructed. The basis vectors are expressed in terms of the Mickelsson lowering operators. Explicit formulas for the matrix elements of generators of ¤(2n) in this basis are given. The basis is natural from the viewpoint of the representation theory of the Yangians. The key role in the construction is played by the fact that the subspace of ¤(2nm 2) highest vectors in any finite-dimensional irreducible representation of ¤(2n) admits a natural structure of a representation of the Yangian Y(‹•(2)).     

李变换方法在超环面光栅成像理论中的应用

应用李变换方法研究了超环面光栅的成像及其像差,介绍了李变换基本原理.光栅成像过程可分成五个部分,分别对应五个李变换;其中最重要的变换就是超环面光栅的衍射李变换.利用这五个李变换推导出了超环面光栅的成像公式,并应用光线追迹,对研究结果进行了验证,说明了李变换的方法能准确地描述超环面光栅的成像.该方法可以处理平面光源,能计算远离子午焦平面处的像差.李变换方法导出的成像公式描述了像平面坐标与物平面坐标、方向余弦之间的函数关系,体现了物空间变量与像空间对应变量之间映射关系.     

On the Structure of Verma Modules over Virasoro and Neveu-Schwarz Algebras

In the paper we present a different proof of the theorem of B. L. Feigin and D. B. Fuchs about the structure of Verma modules over Virasoro algebra. We state some new results about the structure of Verma modules over Neveu-Schwarz. The proof has two advantages: First, it is simpler in the most interesting cases (for example in the so called minimal models), second, it can be generalized for Neveu-Schwarz algebra for some class of Verma modules. Received: 15 December 1995 / Accepted: 4 March 1996     

A Characterization of Affine Kac-Moody Lie Algebras

We give a new characterization of the affine Kac-Moody algebras in terms of extended affine Lie algebras. We also present new realizations of the twisted affine Kac-Moody algebras. Received: 17 April 1996 / Accepted: 11 October 1996     

Highest Weight Modules Over Quantum Queer Superalgebra {U_q(\mathfrak {q}(n))}

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))}. The key ingredients are the triangular decomposition of U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))} and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))}-modules in the category O_q ^{3 0}{\mathcal {O}_{q}^{\geq 0}}.     

Lie Algebras for Constructing Nonlinear Integrable Couplings

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their Hamiltonian structures are also generated. The approach presented in the paper can also
provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.     

Classification of Finite Irreducible Modules over the Lie Conformal Superalgebra CK 6

We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK ₆, for which E(1, 6) is the annihilation superalgebra.     

Formality Theorem for Quantizations of Lie Bialgebras

Using the theory of props we prove a formality theorem associated with universal quantizations of Lie bialgebras.