Lattice Of Submodules Research Articles

ESSENTIAL AND RETRACTABLE GALOIS CONNECTIONS

For bounded lattices, we introduce certain Galois connections, called (cyclically) essential, retractable and UC Galois connections, which behave well with respect to concepts of module-theoretic nature involving essentiality. We show that essential retractable Galois connections preserve uniform dimension, whereas essential retractable UC Galois connections induce a bijective correspondence between sets of closed elements. Our results are applied to suitable Galois connections between submodule lattices. Cyclically essential Galois connections unify semi-projective and semi-injective modules, while retractable Galois connections unify retractable and coretractable modules.

ON THE NOTION OF STRONG IRREDUCIBILITY AND ITS DUAL

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.

On a family of Hopf algebras of dimension 72

We investigate a family of Hopf algebras of dimension 72 whose coradical is isomorphic to the algebra of functions on S_3. We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules. We show that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other.

On Some Geometric Representations of GL N (𝔬)

We study a family of complex representations of the group GL n (𝔬), where 𝔬 is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL n (F) to its maximal compact subgroup GL n (𝔬). We compute explicitly the transition matrix between a geometric basis of the Hecke algebra associated with the representation and an algebraic basis that consists of its minimal idempotents. The transition matrix involves combinatorial invariants of lattices of submodules of finite 𝔬-modules. The idempotents are p-adic analogs of the multivariable Jacobi polynomials.

Generalised Jantzen filtration of Lie superalgebras I

A Jantzen type filtration for generalised Verma modules of Lie superalgebras is introduced. In the case of type I Lie superalgebras, it is shown that the generalised Jantzen filtration for any Kac module is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan–Lusztig polynomials. Furthermore, the length of the Jantzen filtration for any Kac module is determined explicitly in terms of the degree of atypicality of the highest weight. These results are applied to obtain a detailed description of the submodule lattices of Kac modules.

Generalised Verma modules for the orthosymplectic Lie superalgebra [formula omitted

The composition factors with their multiplicities and the submodule lattices are determined for generalised Verma modules over the orthosymplectic Lie superalgebra ospk|2. The results enable us to obtain explicit formulae for the formal characters and dimensions of the finite-dimensional irreducible modules. Applying these results, we also compute the first and second cohomology groups of the Lie superalgebra with coefficients in finite-dimensional Kac modules and irreducible modules.

Metabelian Lie powers of the natural module for a general linear group

Consider a free metabelian Lie algebra M of finite rank r over an infinite field K of prime characteristic p. Given a free generating set, M acquires a grading; its group of graded automorphisms is the general linear group G L r ( K ) , so each homogeneous component M d is a finite dimensional G L r ( K ) -module. The homogeneous component M 1 of degree 1 is the natural module, and the other M d are the metabelian Lie powers of the title. This paper investigates the submodule structure of the M d . In the main result, a composition series is constructed in each M d and the isomorphism types of the composition factors are identified both in terms of highest weights and in terms of Steinbergʼs twisted tensor product theorem; their dimensions are also given. It turns out that the composition factors are pairwise non-isomorphic, from which it follows that the submodule lattice is finite and distributive. By the Birkhoff representation theorem, any such lattice is explicitly recognizable from the poset of its join-irreducible elements. The poset relevant for M d is then determined by exploiting a 1975 paper of Yu.A. Bakhturin on identical relations in metabelian Lie algebras.

On the permutation modules for orthogonal groups [formula omitted] acting on nonsingular points of their standard modules

We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups O m ± ( 3 ) with m ≥ 6 on nonsingular points of their standard modules. These actions, together with those studied in Hall and Nguyen (2011) [2] are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.

Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals

Let M 1 , … , M n be right modules over a ring R . Suppose that the endomorphism ring End R ( M i ) of each module M i has at most two maximal right ideals. Is it true that every direct summand of M 1 ⊕ ⋯ ⊕ M n is a direct sum of modules whose endomorphism rings also have at most two maximal right ideals? We show that the answer is negative in general, but affirmative under further hypotheses. The endomorphism ring of uniserial modules, that is, the modules whose lattice of submodules is linearly ordered under inclusion, always has at most two maximal right ideals, and Pavel Příhoda showed in 2004 that the answer to our question is affirmative for direct sums of finitely many uniserial modules.

Some simple modules for classical groups and p-ranks of orthogonal and Hermitian geometries

We determine the characters of the simple composition factors and the submodule lattices of certain Weyl modules for classical groups. The results have several applications. The simple modules arise in the study of incidence systems in finite geometries and knowledge of their dimensions yields the p-ranks of these incidence systems.

DIRECT SUMS OF INFINITELY MANY KERNELS

AbstractLet 𝒦 be the class of all rightR-modules that are kernels of nonzero homomorphisms φ:E1→E2for some pair of indecomposable injective rightR-modulesE1,E2. In a previous paper, we completely characterized when two direct sumsA1⊕⋯⊕AnandB1⊕⋯⊕Bmof finitely many modulesAiand Bjin 𝒦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, manyAiandBjin 𝒦. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class 𝒦 with the class 𝒰 of all uniserial rightR-modules (a module is uniserial when its lattice of submodules is linearly ordered).

Orthogonal functions generalizing Jack polynomials

The rational Cherednik algebra ℍ is a certain algebra of differential-reflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for ℍ. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Dunkl and Opdam. In this case, the irreducible W-modules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M(λ) consisting of orthogonal functions on ℂ n with values in the representation S λ . For G(1,1, n) with λ = (n) these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of M(λ) in the case in which the orthogonal functions are all well-defined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of ℍ so that the rational Cherednik algebra for G(r,p, n) is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for G(r, p, n) by Clifford theory.

The product of two von Neumann n-frames, its characteristic, and modular fractal lattices

Let L be a bounded lattice. If for each a1 < b1 ∈ L and a2 < b2 ∈ L there is a lattice embedding ψ: [a1, b1] → [a2, b2] with ψ(a1) = a2 and ψ(b1) = b2, then we say that L is a quasifractal. If ψ can always be chosen to be an isomorphism or, equivalently, if L is isomorphic to each of its nontrivial intervals, then L will be called a fractal lattice. For a ring R with 1 let \({\mathcal{V}}(R)\) denote the lattice variety generated by the submodule lattices of R-modules. Varieties of this kind are completely described in [16]. The prime field of characteristic p will be denoted by Fp.

The Frobenius structure of local cohomology

Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an antinilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the ascending chain condition.

Algebraic representation of mappings between submodule lattices

We show that under certain weak conditions on the module R M, every mapping $$ f:\mathfrak{L}\left( {_R M} \right) \to \mathfrak{L}\left( {_S N} \right) $$ between the submodule lattices which preserves arbitrary joins and “disjointness” has a unique representation of the form f(u) = 〈h[ S B R × R U]〉 for all u ∈ $$ \mathfrak{L}\left( {_R M} \right) $$, where S B R is some bimodule and h is an R-balanced mapping. Furthermore, f is a lattice homomorphism if and only if B R is flat and the induced S-module homomorphism $$ \bar h:_S B \otimes _R M \to _S N $$ is monic. If S N also satisfies the same weak conditions, then f is a lattice isomorphism if and only if B R is a finitely generated projective generator, S ≅ End(B R ) canonically, and $$ \bar h:_S B \otimes _R M \to _S N $$ is an S-module isomorphism, i.e., every lattice isomorphism is induced by a Morita equivalence between R and S and a module isomorphism.

A Notion of Zero Dynamics for Linear, Time-delay System

The aim of this paper is to introduce a notion of zero dynamics for linear, time-delay systems. To this aim, we use the correspondence between time-delay systems and system with coefficients in a ring, so to exploit algebraic and geometric methods. By combining the algebraic notion of zero module and the geometric structure of the lattice of invariant submodules of the state module, we point out a natural way to define zero dynamics in particular situations. Then, we extend our approach, in order to encompass more general situations, and we provide a definition of zero dynamics which applies to a number of interesting cases, including that of time-delay systems with commensurable delays. Relations between this notion and fixed poles in closed loop control schemes, as well as with a concept of phase minimality, are discussed.

Honest submodules

Lattices of submodules of modules and the operators we can define on these lattices are useful tools in the study of rings and modules and their properties. Here we shall consider some submodule operators defined by sets of left ideals. First we focus our attention on the relationship between properties of a set of ideals and properties of a submodule operator it defines. Our second goal will be to apply these results to the study of the structure of certain classes of rings and modules. In particular some applications to the study and the structure theory of torsion modules are provided.

Distributive congruence lattices of congruence-permutable algebras

We prove that every distributive algebraic lattice with at most ℵ 1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The ℵ 1 bound is optimal, as we find a distributive algebraic lattice D with ℵ 2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither of any group nor of any module), thus solving negatively a problem of E.T. Schmidt from 1969. Furthermore, D may be taken as the congruence lattice of the free bounded lattice on ℵ 2 generators in any non-distributive lattice variety. Some of our results are obtained via a functorial approach of the semilattice-valued ‘distances’ used by B. Jónsson in his proof of Whitman's Embedding Theorem. In particular, the semilattice of compact elements of D is not the range of any distance satisfying the V-condition of type 3/2. On the other hand, every distributive 〈 ∨ , 0 〉 -semilattice is the range of a distance satisfying the V-condition of type 2. This can be done via a functorial construction.

Elementary equivalence of categories of modules over rings, endomorphism rings, and automorphism groups of modules

In this paper we give a brief review of some recent results of elementary equivalence of linear and algebraic groups and our last new results of elementary equivalence of categories of modules, endomorphism rings of modules, lattices of submodules of modules, and automorphism groups of modules.

Rank 3 permutation modules of the finite classical groups

The cross-characteristic permutation modules for the actions of the finite classical groups on singular 1-spaces of their natural modules are studied. The composition factors and submodule lattices are determined.