A Class of Simple Modules for Electrical Lie Algebra of Type D_(5)

The simple modules for electrical Lie algebra of type D5 were investigated.The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vectors of the Verma modules.The simple highest weight module is isomorphic to either that for the symplectic Lie algebra sp4 or Verma module.

Construction of Simple Modules over the Quantum Affine Space

The coordinate ring O_(q)(K^(n))of quantum affine space is the K-algebra presented by generators x_(1),...,x_(n) and relations x_(i)x_(j)=q_(ij)a_(j)a_(i) for all i,j.We construct simple O_(q)(K^(n))-modules in a more general setting where the parameters qij lie in a torsion subgroup of K^(*)and show that analogous results hold as in the uniparameter case.

Derivations of the Schrodinger Algebra on Restricted Simple Modules

Over an algebraically closed field of characteristic p>2,based on the results on the representation theory of special linear Lie algebra sl(2),restricted simple modules L(λ) of the Schrodinger algebra S(1)are determined,and all derivations of S(1)on L(λ)are also obtained.As an application,the first cohomology of S(1)with the coefficient in L(λ)is determined.

The First Cohomology of osp(1,2)with Coefficients in Baby Verma Modules and Simple Modules

Over a field of characteristic p>0,the first cohomology of the orthogonal symplectic Lie superalgebra osp(1,2)with coefficients in baby Verma modules and simple modules is determined by use of the weight space decompositions of these modules relative to a Cartan subalgebra of osp(1,2).As a byproduct,the first cohomology of osp(1,2)with coefficients in the restricted enveloping algebra(under the adjoint action)is not trivial.

Module Structures on U(ε^(-))for the Euclidean Algebra

We classify modules which are free of rank 1 when restricted to the negative part for the Euclidean Lie algebra.Sufficient and necessary conditions for the simplicity of these modules are given.From these modules we construct modules which are infinitely generated when restricted to the maximal toral subalgebra.

Simple Singular Whittaker Modules Over the Schrödinger Algebra

There are no simple singular Whittaker modules over most of important algebras,such as simple complex finite-dimensional Lie algebras,affine Kac-Moody Lie algebras,the Virasoro algebra,the Heisenberg-Virasoro algebra and the Schrödinger-Witt algebra.In this paper,however,we construct simple singular Whittaker modules over the Schrödinger algebra.Moreover,simple singular Whittaker modules over the Schrödinger algebra are classified.As a result,simple modules for the Schrödinger algebrawhich are locally finite over the positive part are completely classified.We also give characterizations of simple highestweight modules and simple singular Whittaker modules.

Tensor modules over Witt superalgebras

In this paper,we study the tensor module P■M over the Witt superalgebra W_(m,n)~+(resp.W_(m,n)),where P is a simple module over the Weyl superalgebra K_(m,n)~+(resp.K_(m,n))and M is a simple weight module over the general linear Lie superalgebra gl(m,n).We obtain the necessary and sufficient conditions for P■M to be simple,and determine all the simple subquotients of P■M when it is not simple.All the work leads to the completion of some classification problems on the weight representation theories of W_(m,n)~+and W_(m,n).

Analysis of VCSEL Array Module Using a Simple Microlens Array

A simple microlens array is designed between VCSEL array and fiber array for integration of array module. We increase the optical coupling efficiency from -32.057 dBm to -0.9054 dBm by using our designed microlens array.

Imaginary Modules over the Affine Nappi-Witten Algebra

In this paper,we consider the imaginary highest weight modules and the imaginary Whittaker modules for the affine Nappi-Witten algebra.We show that simple singular imaginary Whittaker modules at level(κ、c)(κ∈C~*)are simple imaginary highest weight modules.The necessary and sufficient conditions for these imaginary modules to be simple are given.All simple imaginary modules are classified.

Whittaker Modules for the Derivation Lie Algebra of Torus with Two Variables

Let L be the derivation Lie algebra of C[t1^±1 , t2^±1 ]. Given a triangle decomposition L = L＋ η ＋ L-, we define a nonsingular Lie algebra homomorphism φ ： L＋ → C and the universal Whittaker L-module We of type φ. We obtain all Whittaker vectors and submodules of We. Moreover, all simple Whittaker L-modules of type φ are determined.

τ-Rigid Modules for Algebras with Radical Square Zero

For a radical square zero algebraΛand an indecomposable rightΛ-module M,whenΛis Gorenstein of finite representation type orτM isτ-rigid,M isτ-rigid if and only if the first two projective terms of a minimal projective resolution of M have no non-zero direct summands in common.In particular,we determine allτ-tilting modules for Nakayama algebras with radical square zero.

A Family of Non-weight Modules over the Virasoro Algebra

In this paper,we consider the tensor product modules of a class of non-weight modules and highest weight modules over the Virasoro algebra.We determine the necessary and sufficient conditions for such modules to be simple and the isomorphism classes among all these modules.Finally,we prove that these simple non-weight modules are new if the highest weight module over the Virasoro algebra is non-trivial.

Restricted Kac modules for special contact Lie superalgebras of odd type

We consider the simple restricted modules for special contact Lie superalgebras of odd type over an algebraically closed field of characteristic p>3.We give a sufficient and necessary condition in terms of typical or atypical weights for restricted Kac modules to be simple.In the process,we also determine the socle for each restricted Kac module and the length for each simple restricted module.

Conjugation in Representations of the Zassenhaus Algebra

Let F be an algebracially closed field of characteristic p】2, and L be the pn-dimensional Zassenhaus algebra with the maximal invariant subalgebra L0 and the standard filtration {Li}|i=-1pn-2. Then the number of isomorphism classes of simple L-modules is equal to that of simple L0-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than pn.