We study the category M consisting of U(sln+1)-modules whose restriction to U(h) is free of rank 1, in particular we classify isomorphism classes of objects in M and determine their submodule ...structure. This leads to new sln+1-modules. For n=1 we also find the central characters and derive an explicit formula for taking tensor product with a simple finite dimensional module.
In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element h and obtain families of ...such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity.
We study modules of certain string algebras, which are referred to as of affine type C˜. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten ...quivers of the string algebras and τ-locally free modules defined by Geiss-Lerclerc-Schröer. In particular, we show that an indecomposable module is τ-locally free if and only if it is preprojective, or preinjective or regular in a tube. As an application, we prove Geiss-Leclerc-Schröer's conjecture on the correspondence between positive roots of type C˜ and τ-locally free modules of the corresponding string algebras. Furthermore, given a positive root α, we show that if α is real, then there is a unique τ-locally free module M (up to isomorphism) with rank_M=α; otherwise there are families of τ-locally free modules with rank_M=α.
Composite-database micro-expression recognition is attracting increasing attention as it is more practical for real-world applications. Though the composite database provides more sample diversity ...for learning good representation models, the important subtle dynamics are prone to disappearing in the domain shift such that the models greatly degrade their performance, especially for deep models. In this paper, we analyze the influence of learning complexity, including input complexity and model complexity, and discover that the lower-resolution input data and shallower-architecture model are helpful to ease the degradation of deep models in composite-database task. Based on this, we propose a recurrent convolutional network (RCN) to explore the shallower-architecture and lower-resolution input data, shrinking model and input complexities simultaneously. Furthermore, we develop three parameter-free modules (i.e., wide expansion, shortcut connection and attention unit) to integrate with RCN without increasing any learnable parameters. These three modules can enhance the representation ability in various perspectives while preserving not-very-deep architecture for lower-resolution data. Besides, three modules can further be combined by an automatic strategy (a neural architecture search strategy) and the searched architecture becomes more robust. Extensive experiments on the MEGC2019 dataset (composited of existing SMIC, CASME II and SAMM datasets) have verified the influence of learning complexity and shown that RCNs with three modules and the searched combination outperform the state-of-the-art approaches.
This research aims to give the decompositions of a finitely generated module over some special ring, such as the principal ideal domain and Dedekind domain. One of the main problems with module ...theory is to analyze the objects of the module. This research was using a literature study on finitely generated modules topics from scientific journals, especially those related to the module theory. And by selective cases we find a pattern to build a conjecture or a hypothesis, by deductive proof, we prove the conjecture and state it as a theorem. The main result in this study is the decomposition of the finitely generated module is a direct sum of the torsion submodule and torsion-free submodule. Since the torsion-free module is always a free module over a principal ideal domain, then the torsion-free submodule is a free module. Last, we generalize the ring, from a principal ideal domain, to a Dedekind domain. We found then the torsion-free submodule became a projective module. Then the decomposition of the finitely generated module is a direct sum of the torsion submodule and the projective submodule. These results should help the researchers to analyze the objects on module theory.
For a smooth affine algebra R of dimension d≥3 over a field k and an invertible alternating matrix χ of rank 2n, the group Sp(χ) of invertible matrices of rank 2n over R which are symplectic with ...respect to χ acts on the right on the set Um2n(R) of unimodular rows of length 2n over R. In this paper, we prove that Sp(χ) acts transitively on Um2n(R) if k is algebraically closed, d!∈k× and 2n≥d.
In this paper, we study the conditions under which an automorphism of a dense subring of the endomorphism ring of a free module is induced by a semilinear isomorphism of the module. This leads us to ...obtain some known results from the works of Bezushchak (2022) 7, Courtemanche and Dugas (2016) 6, Wolfson (1962) 8 and Jacobson (1975) 5, along with several nontrivial generalizations.
The formalism of injective stabilization of additive functors is used to define a new notion of the torsion submodule of a module. It applies to arbitrary modules over arbitrary rings. For arbitrary ...modules over commutative domains it coincides with the classical torsion, and for finitely presented modules over arbitrary rings it coincides with the Bass torsion. A formally dual approach – based on projective stabilization – gives rise to a new concept: the cotorsion quotient module of a module. This is done in complete generality – the new concept is defined for any module over any ring. Unlike torsion, cotorsion does not have classical prototypes.
General properties of these constructs are established. It is shown that the Auslander-Gruson-Jensen functor applied to the cotorsion functor returns the torsion functor. As a consequence, a ring is one-sided absolutely pure if and only if each pure injective on the other side is cotorsion-free. If the injective envelope of the ring is finitely presented, then the right adjoint of the Auslander-Gruson-Jensen functor applied to the torsion functor returns the cotorsion functor. This correspondence establishes a duality between torsion and cotorsion over such rings. In particular, this duality applies to artin algebras. It is also shown that, over any ring, the character module of the torsion of a module is isomorphic to the cotorsion of the character module of the module. Under various finiteness conditions on the injective envelope of the ring, the derived functors of torsion and cotorsion are computed.
Let g be a complex simple Lie algebra and Z(g) be the center of the universal enveloping algebra U(g). Denote by Vλ the finite-dimensional irreducible g-module with highest weight λ. Lehrer and Zhang ...defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra EndU(g)(Vλ⊗r) in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the g-invariant endomorphism algebras Rλ(g)=(EndVλ⊗U(g))g and Rλ,π(g)=(EndVλ⊗π(U(g)))g. In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs (g,Vλ), which are multiplicity free and for such pairs, Rλ(g) and Rλ,π(g) are generated by generalizations of the quadratic Casimir elements of Z(g).
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