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大学・研究所にある論文を検索できる 「QUANTUM INVARIANTS OF FRAMED LINKS FROM TERNARY SELF-DISTRIBUTIVE COHOMOLOGY」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

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QUANTUM INVARIANTS OF FRAMED LINKS FROM TERNARY SELF-DISTRIBUTIVE COHOMOLOGY

Zappala, Emanuele 大阪大学 DOI:10.18910/89329

2022.10

概要

The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter, Jelsovsky, Kamada, Langford and Saito in Transactions of the American Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant. In this article we show that the ribbon cocycle invariant is a quantum invariant. We do so by constructing a ribbon category from a TSD set whose twisting and braiding morphisms entail a given TSD 2-cocycle. Then we show that the quantum invariant naturally associated to this braided category coincides with the cocycle invariant. We generalize this construction to symmetric monoidal categories and provide classes of examples obtained from Hopf monoids and Lie algebras. We further introduce examples from Hopf-Frobenius algebras, objects studied in quantum computing.

参考文献

[1] N. Andruskiewitsch and H.J. Schneider: On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math. (2) 171 (2010), 375–417.

[2] Y. Bespalov, T. Kerler, V. Lyubashenko and V. Turaev: Integrals for braided Hopf algebras, J. Pure and Appl. Algebra 148 (2000), 113–164.

[3] F. Bonchi, P. Soboci´nski and F. Zanasi: Interacting Hopf algebras, J. Pure Appl. Algebra 221 (2017), 144–184.

[4] J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito: Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), 3947–3989.

[5] J.S. Carter, M. Elhamdadi, M. Gra˜na and M. Saito: Cocycle knot invariants from quandle modules and generalized quandle homology, Osaka J. Math. 42 (2005), 499–541.

[6] J. Collins and R. Duncan: Hopf-Frobenius algebras and a simpler Drinfeld double; in Proceedings 16th International Conference on Quantum Physics and Logic 2019 (QPL), Electron. Proc. Theor. Comput. Sci. 318, EPTCS, 2020, 150–180.

[7] M. Eisermann: Yang–Baxter deformations of quandles and racks, Algebr. Geom. Topol. 5 (2005), 537–562.

[8] M. Elhamdadi, M. Saito and E. Zappala: Higher Arity Self-Distributive Operations in Cascades and their Cohomology, J. Algebra Appl. 20 (2021), Paper No. 2150116, 33pp.

[9] M. Elhamdadi, M. Saito and E. Zappala: Heap and Ternary Self-Distributive Cohomology, Comm. Algebra 49 (2021), 2378–2401.

[10] P. Etingof and S. Gelaki: Invariant Hopf 2-cocycles for affine algebraic groups, Int. Math. Res. Not. 2020 (2020), 344–366.

[11] A. Ishii, M. Iwakiri, Y. Jang and K. Oshiro: A G-family of quandles and handlebody-knots, Illinois J. Math. 57 (2013), 817–838.

[12] P.J. Freyd and D.N. Yetter: Braided compact closed categories with applications to low dimensional topology, Adva. Math. 77 (1989), 156–182.

[13] M. Gra˜na: Quandle knot invariants are quantum knot invariants, J. Knot Theory Ramifications 11 (2002), 673–681.

[14] M.J. Green: Generalizations of Quandles and their cohomologies, PhD dissertation, University of South Florida, 2018.

[15] D. Joyce: A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), 37–65.

[16] C. Kassel: Quantum Groups, Springer Science & Business Media, 2012.

[17] A.Klimyk and K. Schm¨udgen: Quantum Groups and Their Representations, Springer Science & Business Media, 2012.

[18] K.H. Ko and L. Smolinsky: The framed braid group and 3-manifolds, Proc. Amer. Mathe. Soc. 115 (1992), 541–551.

[19] R.G. Larson and M.E. Sweedler: An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75–94.

[20] S.V. Matveev: Distributive groupoids in knot theory, Mat. Sb. 119 (161) (1982), 78–88.

[21] T. Nosaka: Quandle cocycles from invariant theory, Adv. Math. 245 (2013), 423–438.

[22] B. Pareigis: When Hopf algebras are Frobenius algebras, J. Algebra 18 (1971), 588–596.

[23] N.Yu. Reshetikhin and V.G. Turaev: Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1–26.

[24] M. Saito and E. Zappala: Fundamental heap for framed links and ribbon cocycle invariants, arXiv:2011.03684 (2020).

[25] M. Saito and E. Zappala: Braided Frobenius Algebras from certain Hopf Algebras, J. Algebra Appl. online ready, available at https://doi.org/10.1142/S0219498823500123.

[26] V.G. Turaev: The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527–554.

[27] E. Zappala: Non-Associative Algebraic Structures in Knot Theory, Ph.D. dissertation, University of South Florida, 2020.

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