Geodesic
Geometric construction of spinors in orthogonal modular categories
Beliakova, Anna1
1Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 969-992
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A geometric construction of ℤ2–graded odd and even orthogonal modular categories is given. Their 0–graded parts coincide with categories previously obtained by Blanchet and the author from the category of tangles modulo the Kauffman skein relations. Quantum dimensions and twist coefficients of 1–graded simple objects (spinors) are calculated. We show that invariants coming from our odd and even orthogonal modular categories admit spin and ℤ2–cohomological refinements, respectively. The relation with the quantum group approach is discussed.

DOI : 10.2140/agt.2003.3.969
Keywords: modular category, quantum invariant, Vassiliev–Kontsevich invariant, weight system

Beliakova, Anna  1

1 Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland 
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Beliakova, Anna. Geometric construction of spinors in orthogonal modular categories. Algebraic and Geometric Topology, Tome 3 (2003) no. 2, pp. 969-992. doi: 10.2140/agt.2003.3.969

[1] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423

[2] D Bar-Natan, T T Q Le, D P Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. 7 (2003) 1

[3] A Beliakova, Refined invariants and TQFTs from Homfly skein theory, J. Knot Theory Ramifications 8 (1999) 569

[4] A Beliakova, C Blanchet, Modular categories of types $B$, $C$ and $D$, Comment. Math. Helv. 76 (2001) 467

[5] A Beliakova, C Blanchet, Skein construction of idempotents in Birman–Murakami–Wenzl algebras, Math. Ann. 321 (2001) 347

[6] C Blanchet, A spin decomposition of the Verlinde formulas for type A modular categories, Comm. Math. Phys. 257 (2005) 1

[7] W Fulton, J Harris, Representation theory, Graduate Texts in Mathematics 129, Springer (1991)

[8] T T Q Le, Quantum invariants of 3–manifolds: integrality, splitting, and perturbative expansion, from: "Proceedings of the Pacific Institute for the Mathematical Sciences Workshop “Invariants of Three–Manifolds” (Calgary, AB, 1999)" (2003) 125

[9] T T Q Le, Integrality and symmetry of quantum link invariants, Duke Math. J. 102 (2000) 273

[10] T Q T Le, J Murakami, The universal Vassiliev–Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996) 41

[11] T T Q Le, J Murakami, Kontsevich's integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996) 39

[12] T Le, V Turaev, Quantum groups and ribbon $G$–categories, J. Pure Appl. Algebra 178 (2003) 169

[13] B Patureau-Mirand, Link invariant for the spinor representation of $\mathfrak{so}_7$, J. Knot Theory Ramifications 12 (2003) 605

[14] V G Turaev, Quantum invariants of knots and 3–manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter Co. (1994)

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