$\mathcal{U}(\mathfrak{h})$-finite modules and weight modules I: weighting functors, almost-coherent families and category $\mathfrak{A}^{\text{irr}}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913833249931264 |
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| author | Mendonça, Eduardo M. |
| author_facet | Mendonça, Eduardo M. |
| contents | This paper builds upon J. Nilsson's classification of rank one $\mathcal{U}(\mathfrak{h})$-free modules by extending the analysis to modules without rank restrictions, focusing on the category $\mathfrak{A}$ of $\mathcal{U}(\mathfrak{h})$-finite $\mathfrak{g}$-modules. A deeper investigation of the weighting functor $\mathcal{W}$ and its left derived functors, $\mathcal{W}_*$, led to the proof that simple $\mathcal{U}(\mathfrak{h})$-finite modules of infinite dimension are $\mathcal{U}(\mathfrak{h})$-torsion free. Furthermore, it is shown that these modules are $\mathcal{U}(\mathfrak{h})$-free if they possess non-integral or singular central characters. It is concluded that the existence of $\mathcal{U}(\mathfrak{h})$-torsion-free $\mathfrak{g}$-modules is restricted to Lie algebras of types A and C.
The concept of an almost-coherent family, which generalizes O. Mathieu's definition of coherent families, is introduced. It is proved that $\mathcal{W}(M)$, for a $\mathcal{U}(\mathfrak{h})$-torsion-free module $M$, falls within this class of weight modules. Furthermore, a notion of almost-equivalence is defined to establish a connection between irreducible semi-simple almost-coherent families and O. Mathieu's original classification.
Progress is also made in classifying simple modules within the category $\mathfrak{A}^{\text{irr}}$, which consists of $\mathcal{U}(\mathfrak{h})$-finite modules $M$ with the property that $\mathcal{W}(M)$ is an irreducible almost-coherent family. A complete classification is achieved for type C, with partial classification for type A. Finally, a conjecture is presented asserting that all simple $\mathfrak{sl}(n+1)$-modules in $\mathfrak{A}^{\text{irr}}$ are isomorphic to simple subquotients of exponential tensor modules, and supporting results are proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathcal{U}(\mathfrak{h})$-finite modules and weight modules I: weighting functors, almost-coherent families and category $\mathfrak{A}^{\text{irr}}$ Mendonça, Eduardo M. Representation Theory This paper builds upon J. Nilsson's classification of rank one $\mathcal{U}(\mathfrak{h})$-free modules by extending the analysis to modules without rank restrictions, focusing on the category $\mathfrak{A}$ of $\mathcal{U}(\mathfrak{h})$-finite $\mathfrak{g}$-modules. A deeper investigation of the weighting functor $\mathcal{W}$ and its left derived functors, $\mathcal{W}_*$, led to the proof that simple $\mathcal{U}(\mathfrak{h})$-finite modules of infinite dimension are $\mathcal{U}(\mathfrak{h})$-torsion free. Furthermore, it is shown that these modules are $\mathcal{U}(\mathfrak{h})$-free if they possess non-integral or singular central characters. It is concluded that the existence of $\mathcal{U}(\mathfrak{h})$-torsion-free $\mathfrak{g}$-modules is restricted to Lie algebras of types A and C. The concept of an almost-coherent family, which generalizes O. Mathieu's definition of coherent families, is introduced. It is proved that $\mathcal{W}(M)$, for a $\mathcal{U}(\mathfrak{h})$-torsion-free module $M$, falls within this class of weight modules. Furthermore, a notion of almost-equivalence is defined to establish a connection between irreducible semi-simple almost-coherent families and O. Mathieu's original classification. Progress is also made in classifying simple modules within the category $\mathfrak{A}^{\text{irr}}$, which consists of $\mathcal{U}(\mathfrak{h})$-finite modules $M$ with the property that $\mathcal{W}(M)$ is an irreducible almost-coherent family. A complete classification is achieved for type C, with partial classification for type A. Finally, a conjecture is presented asserting that all simple $\mathfrak{sl}(n+1)$-modules in $\mathfrak{A}^{\text{irr}}$ are isomorphic to simple subquotients of exponential tensor modules, and supporting results are proved. |
| title | $\mathcal{U}(\mathfrak{h})$-finite modules and weight modules I: weighting functors, almost-coherent families and category $\mathfrak{A}^{\text{irr}}$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2411.18390 |