{
  "abstract": "This paper introduces a Lie algebra analogue of the middle convolution functor, which is defined on the category of modules over certain Lie algebras, including, as typical motivating examples, free Lie algebras, Drinfeld-Kohno Lie algebras, and the holonomy Lie algebras of complements of hyperplane arrangements. First, we demonstrate that the middle convolution for Lie algebra representations can be regarded as a natural generalization of the infinitesimal analogue of the Long-Moody functor for Drinfeld-Kohno Lie algebras. Second, we show that our middle convolution recovers the Dettweiler-Reiter additive middle convolution for Fuchsian systems on the punctured Riemann sphere as a special case. Furthermore, we show that when applied to the holonomy Lie algebra of the complement of a hyperplane arrangement, our functor is compatible with Haraoka's middle convolution for logarithmic connections on such complements. Finally, we establish a Riemann-Hilbert correspondence between the middle convolution for the holonomy Lie algebra and the middle convolution for local systems on complements of hyperplane arrangements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.09828",
  "author_total": 1,
  "authors": [
    {
      "name": "Kazuki Hiroe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5fpwgaecpsk2f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.09828",
  "fulltext_url": null,
  "id": "5fpwgaecpsk2f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.09828",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7160922524",
  "pdf_url": "https://arxiv.org/pdf/2605.09828",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5fpwgaecpsk2f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2605.09828",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.09828"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Middle convolution for Lie algebra representations",
  "title_text": "Middle convolution for Lie algebra representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/5fpwgaecpsk2f",
  "venue": "arXiv",
  "versions": []
}