Finite-dimensional Representations for a Class of Generalized Intersection Matrix Lie Algebras

In this paper, the authors study a class of generalized intersection matrix Lie algebras gim(M_n), and prove that its every finite-dimensional semisimple quotient is of type M(n, a, c, d). Particularly, any finite dimensional irreducible gim(M_n) module must be an irreducible module of Lie algebra of type M(n, a, c, d) and any finite dimensional irreducible module of Lie algebra of type M(n, a, c, d) must be an irreducible module of gim(M_n).     

The eigenvalues of four-dimensional rotations

Let X be an orthosymplectie Lie superalgebra of type B or D. The weight structure of the tensor product moduleW=? ^M V, of M-copies of the natural representation. V, of X is studied from a partition point of view. A combinatorial characterization of the dominant weights of W and the weights of W which are highest weights for the finite dimensional irreducible modules is given. This partition point of view allows us to prove that the dominant weights of W and the weights of W which are highest weights for the finite dimensional irreducible modules stabilize as the rank of X gets large.     

Strongly Irreducible Submodules of Modules

Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.     

Irreducible weight modules for the Virasoro-like algebra and its q-analog

In this paper, using Larsson’s functor with irreducible 𝔰𝔩₂-modules V, we construct a class of ?²-graded modules for the Virasoro-like algebra and its q-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.     

On Whittaker modules over a class of algebras similar to U(sl 2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A ₁ and finding interesting families of new noetherian rings, a class of algebras similar to U(sl ₂) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.      

The Stable Class of the Augmentation Ideal

In [F.E.A. Johnson, Stable Modules and the D(2)-Problem, LMS Lecture Notes In Mathematics, vol. 301, CUP (2003)], for finite groups G, we gave a parametrization of the stable class of the augmentation ideal of Z[G] in terms of stably free modules. Whilst the details of this parametrization break down immediately for infinite groups, nevertheless one may hope to find parallel arguments for restricted classes of infinite groups. Subject to the restriction that Ext¹(Z, Z[G]) = 0, we parametrize the minimal level in Ω₁(Z) by means of stably free modules and give a lower estimate for the size of Ω₁(Z).     

A Generalization of Lie's Theorem

We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L.     

Gelfand-Kirillov dimensions of the ℤ-graded oscillator representations of 𝔬(n,ℂ) and 𝔰𝔭(2n,ℂ)

In this paper, we give a method to compute the Gelfand–Kirillov dimensions of some polynomial–type weight modules. These modules are infinite-dimensional irreducible 𝔬(n,?)-modules and 𝔰𝔭(2n,?)-modules that appeared in the ?-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules.     

On Characteristic Modules of Graded Quasi-hereditary Algebras

Abstract

Let G be a group and A a G-graded quasi-hereditary algebra. Then its characteristic module is proved to be G-gradable, i.e., it is isomorphic to a G-graded module as A-modules. This implies that the Ringel dual A′ of A admits a canonical G-grading which extends to the graded situation the typical equivalence between Δ-good and ?-good modules of A and A′, respectively. It follows some consequences: the derived category of finitely generated G-graded A-modules is equivalent to the derived category of finitely generated G-graded A'-modules; if G is finite, then the Ringel dual of the smash product A#G* is isomorphic to the smash product A'#G* of A' with G.     

On M-coidempotent elements and fully coidempotent modules

Abstract

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then M is fully coidempotent if and only if M is a vn-regular module. Finally, we give a characterization of semisimple Artinian modules via weak idempotents.     

一类超W-代数权空间维数有限的不可约模

本文研究了一类超W-代数上某一权空间维数有限的不可约权模,证明了该权模必是Harish-Chandra模.     

A Twining Character Formula For Demazure Modules

We prove a formula for the twining characters of certain Demazure modules, over a Borel subalgebra \mathfrakb\mathfrak{b} of a finite dimensional complex semisimple Lie algebra \mathfrakg\mathfrak{g}. This formula describes the twining character of the Demazure module by the w\omega-Demazure operator associated to an element of the Weyl group that is fixed by the Dynkin diagram automorphism w\omega of \mathfrakg\mathfrak{g}. Our result is a refinement of the twining character formula for the irreducible highest weight \mathfrakg\mathfrak{g}-modules of symmetric dominant integral highest weights, and also of the ordinary Demazure character formula.     

Unicity of best one-sided L1-approximations for certain classes of spline functions

The chief purpose of this paper is to present conditions ensuring uniqueness of best one-sided L¹-approximations for approximation by finite dimensional subspaces of differentiable functions. Using these results it is shown that uniqueness of such best approximations will hold for a class of generalized spline subspaces and also for spline subspaces satisfying certain boundary conditions. These considerations have an important application to uniqueness of quadrature formulae of “highest possible degree of precision”.     

Regularity of vertex operator superalgebras

We prove that the regularity of a vertex operator superalgebra can be reduced to the semisimplicity of the category of its weak modules.Moreover,the rationality can be replaced by requiring that each 1/2Z+-graded module is a direct sum of irreducible weak modules.     

Special bases for solutions of a generalized Maxwell system in 3‐dimensional space

The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L² on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2.     

Ergodicity of the Finite and Infinite Dimensional α-Stable Systems

Abstract

Some finite and infinite dimensional perturbed α-stable dynamics are constructed and studied in this article. We prove that the finite dimensional system is strongly mixing, while in the infinite dimensional case that the functional coercive inequalities are not available, we develop and apply a technique to prove the point-wise ergodicity for systems with sufficiently small interaction in a large subspace of Ω = R ^{Z d} .     

Nonzero level Harish-Chandra modules over the Virasoro-like algebra

In the present paper, we study the nonzero level Harish-Chandra modules for the Virasoro-like algebra. We prove that a nonzero level Harish-Chandra module of the Virasoro-like algebra is a generalized highest weight (GHW for short) module. Then we prove that a GHW module of the Virasoro-like algebra is induced from an irreducible module of a Heisenberg subalgebra.     

Monomial Modules and Endo-Monomial Modules

We determine the multiplicity algebras and multiplicity modules of a p-monomial module. For a general p-group P, we find a sufficient and necessary condition for an endo-monomial P-module to be an endo-permutation P-module, and prove that a capped indecomposable endo-monomial P-module is of p ^′-rank. At last, we give an alternative definition of the generalized Dade P-group.     

Periodic equivariant realK-theories have rational Tate theory

Summary We study a generalized equivariantK-theory introduced by M. Karoubi. We prove, that it is anRO (G, U)-graded cohomology-theory and that the associated Tate spectrum is rational whenG is finite. This implies that for finite groups, the Atiyah-Segal Real equivariantK-theories have rational Tate theory. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag     

Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.