Quantum determinants in ribbon category

Document Type : Research Paper

Authors

Mathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of Sciences Dhar Al Mahraz, P. O. Box 1796, University Sidi Mohamed Ben Abdellah Fez, Morocco.

Abstract

The aim of this paper is to introduce an abstract notion of determinant which we call quantum determinant, verifying the properties of the classical one. We introduce R−basis and R−solution on rigid objects of a monoidal 𝐴𝑏−category, for a compatibility relation R, such that we require the notion of duality introduced by Joyal and Street, the notion given by Yetter and Freyd and the classical one, then we show that R−solutions over a semisimple ribbon 𝐴𝑏−category form as well a semisimple ribbon 𝐴𝑏−category. This allows us to define a concept of so-called quantum determinant in ribbon category. Moreover, we establish relations between these and the classical determinants. Some properties of the quantum determinants are exhibited.

Keywords

Main Subjects

References

[1] Eilenberg, S. and Kelly, G.M., Closed categories, Proc. Conf. Categorical Algebra at La Jolla (Springer) (1966), 421-562.
[2] Freyd, P. and Yetter, D., Braided compact closed categories with application to low dimensional topology, Adv. Math. 77 (1989), 156-182.
[3] Geer, N., Kujawa, J., and Patureau-Mirand, B., Generalized trace and modified dimension functions on ribbon categories, Sel. Math. New Ser. 17 (2011), 453-504.
[4] Geer, N., Kujawa, J., and Patureau-Mirand, B., M-traces in (non unimodular) pivotal categories, Algebras Represent. Theory (2021).
[5] Geer, N., Patureau-Mirand, B., and Turaev, V., Modified quantum dimensions and renormalized link invariants, Compos. Math. 145(1) (2009), 196-212.
[6] Geer, N., Patureau-Mirand, B., and Virelizer, A., Traces on ideals in pivotal categories, Quantum Topol. 4(1) (2013), 91-124.
[7] Joyal, A. and Street, R., “Braided monoidal categories”, Mathematics Reports 86008, Macquarie University, 1986.
[8] Joyal, A. and Street, R., Braided tensor categories, Adv. Math. 102 (1993), 20-78.
[9] Joyal, A. and Street, R., The geometry of Tensor Calculus (1), Adv. Math. 88 (1991), 55-112.
[10] Kassel, C., “Quantum Groups”, Gradute Texts in Mathematics 155, Springer-Verlag, 1995.
[11] Kelly, G. and Laplaza, M., Coherence for compact closed categories, J. Pure Appl. algebra 19 (1980), 193-213.
[12] Mac Lane, S., “Categories for the Working Mathematician”, Graduate Texts in Mathematics 5, Springer-Verlag, 1971.
[13] Ngoc Phu, H. and Huyen Trang, N., Generalisation of traces in pivotal categories, J. Sci. Technol. 17(4) (2019), 20-29.
[14] Reshetikhin, N.Y. and Turaev, V.G., Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547-598.
[15] Shum, M.C. Tortile tensor categories, J. Pure Appl. Algebra 93 (1994), 57-110.
[16] Street, R., Ideals, radicals, and structure of additive categories, Appl. Categor. Struct. 3 (1995), 139-149.
[17] Turaev, V.G., Modular categories and 3-manifolds invariants, Int. J. Mod. Phys. B, 6, Nos. 11-12 (1992), 1807-1824.
[18] Turaev, V.G. and Virelizier, A., “Monoidal Categories and Topological Field Theory”, Progress in Mathematics 322, Birkhuser/Springer, 2017.
Categories and General Algebraic Structures with Applications
Volume 17, Issue 1
July 2022
Pages 203-232

Files

Share

How to cite

Statistics