Abstract
A quandle is an algebraic system originated in knot theory, and can be regarded as a generalization of symmetric spaces. The inner automorphism group of a quandle is defined as the group generated by the point symmetries (right multiplications). In this paper, starting from any simple graphs, we construct quandles whose inner automorphism groups are abelian. We also prove that the constructed quandle is homogeneous if and only if the graph is vertex-transitive. This shows that there is a wide family of quandles with abelian inner automorphism groups, even if we impose the homogeneity. The key examples of such quandles are realized as subquandles of oriented real Grassmannian manifolds.
Keywords
Quandles symmetric spaces inner automorphism groups vertex-transitive graphs real Grassmannian manifolds
Thanks
This work was supported by JSPS KAKENHI Grant Numbers JP19K21831, JP22H01124. This work was partly supported by MEXT Promotion of Distinctive Joint Research Center Program JPMXP0723833165.
References
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- [21] Wada, K.: Two-point homogeneous quandles with cardinality of prime power. Hiroshima Math. J. 45, 165–174 (2015).
Abstract
References
- [1] Bonatto, M.: Principal and doubly homogeneous quandles, Monatsh. Math. 191, 691–717 (2020).
- [2] Carter, J. S.: A survey of quandle ideas. In: Introductory lectures on knot theory, Ser. Knots Everything 46, 22–53. World Sci. Publ., Hackensack, NJ (2012).
- [3] Carter, J. S., Elhamdadi, M., Nikiforou, M. A., Saito, M.: Extensions of quandles and cocycle knot invariants, J. Knot Theory Ramifications 12, 725–738 (2003).
- [4] Carter, J. S., Kamada, S., Saito, M.: Diagrammatic computations for quandles and cocycle. In: Diagrammatic Morphisms and Applications, Contemp. Math. 318, 51–74 (2003).
- [5] Chen, B. Y.: Two-numbers and their applications.a survey. Bull. Belg. Math. Soc. Simon Stevin 25, 565–596 (2018).
- [6] Chen, B. Y., Nagano, T.: A Riemannian geometric invariant and its applications to a problem of Borel and Serre. Trans. Amer. Math. Soc. 308, 273–297 (1988).
- [7] Hulpke, A., Stanovský, D., Vojtˇechovský, P.: Connected quandles and transitive groups. J. Pure Appl. Algebra 220, 735–758 (2016).
- [8] Jedlicka, P., Pilitowska, A., Stanovský, D., Zamojska-Dzienio, A.: The structure of medial quandles. J. Algebra 443, 300–334 (2015).
- [9] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Graduate Studies in Mathematics 34, American Mathematical Society, Providence, RI (2001).
- [10] Ishihara, Y., Tamaru, H.: Flat connected finite quandles. Proc. Amer. Math. Soc. 144, 4959–4971 (2016).
- [11] Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23, 37–65 (1982).
- [12] Kamada, S.: Kyokumen musubime riron (Surface-knot theory, in Japanese). Springer Gendai Sugaku Series 16, Maruzen Publishing Co. Ltd (2012).
- [13] Kamada, S., Tamaru, H., Wada, K.: On classification of quandles of cyclic type. Tokyo J. Math. 39, 157–171 (2016).
- [14] Kubo, K., Nagashiki, M., Okuda, T., Tamaru, H.: A commutativity condition for subsets in quandles — a generalization of antipodal subsets. In: Differential Geometry and Global Analysis; In Honor of Tadashi Nagano, Contemp. Math. 777, 103–125 (2022).
- [15] Loos, O.: Symmetric spaces I: General theory. W. A. Soc. Benjamin, Inc., New York-Amsterdam (1969).
- [16] Matveev, S. V.: Distributive groupoids in knot theory. Math. USSR-Sb. 47, 73–83 (1984).
- [17] Singh, M.: Classification of flat connected quandles. J. Knot Theory Ramifications 25, 1650071 [8 pages] (2016).
- [18] Stanovský, D.: The origins of involutory quandles. ArXiv:1506.02389 (2015).
- [19] Tamaru, H.: Two-point homogeneous quandles with prime cardinally. J. Math. Soc. Japan 65, 1117–1134 (2013).
- [20] Vendramin, L.: Doubly transitive groups and cyclic quandles. J. Math. Soc. Japan 69, 1051–1057 (2017).
- [21] Wada, K.: Two-point homogeneous quandles with cardinality of prime power. Hiroshima Math. J. 45, 165–174 (2015).
Details
| Primary Language | English |
|---|---|
| Subjects | Algebraic and Differential Geometry |
| Journal Section | Research Article |
| Authors | |
| Early Pub Date | April 6, 2024 |
| Publication Date | April 23, 2024 |
| Submission Date | February 17, 2024 |
| Acceptance Date | March 28, 2024 |
| Published in Issue | Year 2024 Volume: 17 Issue: 1 |
