研究者業績

軽尾 浩晃

カルオ ヒロアキ  (Hiroaki Karuo)

基本情報

所属
学習院大学 理学部 数学科 助教
学位
理学(2022年3月 京都大学)

連絡先
hiroaki.karuogakushuin.ac.jp
研究者番号
80963363
ORCID ID
 https://orcid.org/0000-0003-4654-2895
J-GLOBAL ID
202201007240998950
researchmap会員ID
R000033304

外部リンク

研究キーワード

 3

論文

 11
  • Hiroaki Karuo, Han-Bom Moon, Helen Wong
    2025年1月18日  
    We consider a generalization of the Kauffman bracket skein algebra of a surface that is generated by loops and arcs between marked points on the interior or boundary, up to skein relations defined by Muller and Roger-Yang. We compute the center of this Muller-Roger-Yang skein algebra and show that it is almost Azumaya when the quantum parameter $q$ is a primitive $n$-th root of unity with odd $n$. We also discuss the implications on the representation theory of the Muller-Roger-Yang generalized skein algebra.
  • Hiroaki Karuo, Zhihao Wang
    accepted for publication in Algebraic & Geometric Topology 2025年  査読有り
    We study a generalized Witten's finiteness conjecture for the skein modules of oriented compact 3-manifolds with boundary. We formulate an equivalent version of the generalized finiteness conjecture using handlebodies and 2-handles, and prove the conjecture for some classes with the handlebodies of genus 2 and 3 using the equivalent version.
  • Tsukasa Ishibashi, Hiroaki Karuo
    Communications in Mathematical Physics 405(10) 2024年10月7日  査読有り
    Abstract We generalize the quantum duality map $$\mathbb {I}_{\mathcal {A } }$$ of Allegretti–Kim (Adv Math 306:1164–1208, 2017) for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map $$\mathbb {I}_{\mathcal {X } }$$ on the dual side based on the quantum bracelets basis (Mandel and Qin in arXiv:2301.11101; Thurston in Proc Natl Acad Sci USA 111(27):9725–9732, 2014). Our construction factors through reduced stated skein algebras, based on the quantum trace maps (Lê in Quantum Topol 9(3):591–632, 2018) together with an appropriate way of skein lifting of integral $$\mathcal {A}$$-laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus.
  • Hiroaki Karuo, Zhihao Wang
    arXiv 2024年8月22日  
    In the paper, we show some properties of (reduced) stated SL($n$)-skein algebras related to their centers for essentially bordered pb surfaces, especially their centers, finitely generation over their centers, and their PI-degrees. The proofs are based on the quantum trace maps, embeddings of (reduced) stated SL($n$)-skein algebras into quantum tori appearing in higher Teichmüller theory. Thanks to the Unicity theorem in [BG02, FKBL19], we can understand the representation theory of (reduced) stated SL($n$)-skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access to the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially.
  • Hiroaki Karuo
    Journal of Algebra 647 312-326 2024年6月  査読有り

MISC

 2

講演・口頭発表等

 49

教育業績(担当経験のある科目)

 10

共同研究・競争的資金等の研究課題

 6

その他

 1