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Multi-Sided Braid Type Subfactors

Published online by Cambridge University Press:  20 November 2018

Juliana Erlijman*
Affiliation:
Department of Mathematics University of Regina Regina, Saskatchewan S4S 0A2, e-mail: erlijman@math.uregina.ca
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Abstract

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We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite $\text{I}{{\text{I}}_{1}}$ factor $R$ in $[\text{E1}]$—which arise by considering unitary braid representations with certain properties—to multi-sided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples—where the braid representations are obtained in connection with the representation theory of Lie algebras of types $A,\,B,\,C,\,D$. We also compute the (first) relative commutants.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2001

References

[B] Birman, J., Braids, links and mapping class groups. Ann.Math. Studies 82, Princeton, New Jersey, Princeton Univ. Press. 1974.Google Scholar
[Ch] Choda, M., Index for factors generated by Jones’ two sided sequence of projections. Pacific J. Math. 139 (1989), 116.Google Scholar
[E1] Erlijman, J., New braid subfactors from braid group representations. Trans. Amer.Math. Soc. 350 (1998), 185211.Google Scholar
[E2] Erlijman, J., Two sided braid groups and asymptotic inclusions. Pacific J. Math. (1) 193 (2000), 5778.Google Scholar
[EK] Evans, D. and Kawahigashi, Y., Quantum symmetries on operator algebras. Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, 1998.Google Scholar
[G] Goto, S., Quantum double construction for subfactors arising from periodic commuting squares. preprint, 1996.Google Scholar
[GHJ] Goodman, F., de la Harpe, P. and Jones, V., Coxeter graphs and towers of algebras. 14, Springer-Verlag, MSRI Publications, 1989.Google Scholar
[GW] Goodman, F. and Wenzl, H., Littlewood-Richardson coefficients for the Hecke algebra at roots of unity. Adv. inMath. 82 (1990), 2445.Google Scholar
[O] Ocneanu, A., Chirality of Operator Algebras. Subfactors, Taniguchi Symposium on Operator Algebras,World Scient. 1994, 39–63.Google Scholar
[P] Popa, S., Orthogonal pairs of *-subalgebras of finite Von Neumann algebras. J. Operator Theory 9 (1983), 253268.Google Scholar
[Wa] Wassermann, A., Operator algebras and conformal field theory. Proc. ICMZürich, Birkäuser, 1994.Google Scholar
[W1] Wenzl, H., Hecke algebras of type An and subfactors. Invent.Math. 92 (1988), 349383.Google Scholar
[W2] Wenzl, H., Quantum groups and subfactors of Lie type B, C, and D. Comm. Math. Phys. 133 (1990), 383433.Google Scholar
[W4] Wenzl, H., Braids and invariants of 3-manifolds. Invent.Math. 114 (1993), 235275.Google Scholar
[X] Xu, F., Jones-Wassermann subfactors for disconnected intervals. preprint, QA/9704003.Google Scholar