It is shown that a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V ) is semisimple and each irreducible admissible V -module is ordinary. A contravariant form on a Verma type admissible V -module is constructed and the radical is exactly the maximal proper submodule. As an application the rationality of V + L for any positive definite even lattice is obtained.

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E9). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s pat...

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E9). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s pat...

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E9). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s pat...

We introduced and studied GF-regular modules as a generalization of π-regular rings to modules as well as regular modules (in the sense of Fieldhouse). An R-moduleM is called GF-regular if for each x ∈ M and r ∈ R, there exist t ∈ R and a positive integer n such that rntrnx = rx. The notion of G-pure submodules was introduced to generalize pure submodules and proved that an RmoduleM isGF-regula...

All rings are commutative with 1 6= 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a, b ∈ R and m ∈M and abm ∈ N , then am ∈M -rad(N) or bm ∈M -rad(N) or ...

The blob algebra is a finite-dimensional quotient of the Hecke type B which almost always quasi-hereditary. We construct indecomposable tilting modules for over field characteristic 0 in doubly critical case. Every module maximal highest weight either projective or an extension simple by module. Moreover, every submodule weight. conclude that graded Weyl filtration multiplicities this case are ...

A new submodule clustering method via sparse and lowrank representation for multi-way data is proposed in this paper. Instead of reshaping multi-way data into vectors, this method maintains their natural orders to preserve data intrinsic structures, e.g., image data kept as matrices. To implement clustering, the multi-way data, viewed as tensors, are represented by the proposed tensor sparse an...

Let R be a commutative ring with identity , and M is unitary left R-module”, “A proper submodule E of an R-module called weakly quasi-prime if whenever r, s ∈ R, m M, 0 ≠ rsm implies that rm or sm E”. “We introduce the concept quasi 2-absorbing as generalization submodule”, where r,s,t ∈M 0≠ rstm rtm stm E. we study basic properties 2-absorbing. Furthermore, relationships other classes module a...

The modular multilevel converter (MMC) is receiving extensive research interests in high/medium voltage applications due to its modularity, scalability, reliability, high voltage capability and excellent harmonic performance. The submodule capacitors are usually quite bulky since they have to withstand fundamental frequency voltage fluctuations. To reduce the capacitance of these capacitors, th...