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Hopf algebras of types $U_q(sl_n)'$ and $O_q(SL_n)'$ which give rise to certain invariants of knots, links and 3-manifolds

Author(s): Shlomo Gelaki; Sara Westreich
Journal: Trans. Amer. Math. Soc. 352 (2000), 3821-3836.
MSC (2000): Primary 16W30
Posted: April 18, 2000
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Abstract:

In this paper we determine when Lusztig's $U_q(sl_n)'$ has all the desired properties necessary to define invariants of knots, links and 3-manifolds. Specifically, we determine when it is ribbon, unimodular and factorizable. We also compute the integrals and distinguished elements involved.

References:

[D1]
V. G. Drinfeld, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321-342. MR 91b:16046

[D2]
V. G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 251-258. MR 87h:58080

[GW]
S. Gelaki and S. Westreich, On the quasitriangularity of $U_q(\mathrm{sl}_n)^{\prime}$, J. London Math. Soc. (2) 57 (1998), 105-125. MR 99e:16051

[H]
M. A. Hennings, Invariants of links and 3-manifolds obtained from Hopf algebras, J. London Math. Soc. (2) 54 (1996), 593-624. MR 98g:57011

[J]
M. Jimbo, Quantum groups and the Yang-Baxter equation, to appear.

[K]
L. H. Kauffman, Gauss codes, quantum groups and ribbon Hopf algebras, Rev. Math. Phys. 5 (1993), 735-773. MR 94k:57013

[KR1]
L. H. Kauffman and D. E. Radford, A necessary and sufficient condition for a finite-dimensional Drinfeld double to be a ribbon Hopf algebra, J. of Algebra 159 (1993), 98-114. MR 94d:16037

[KR2]
L. H. Kauffman and D. E. Radford, Invariants of 3-manifolds derived from finite dimensional Hopf algebras, J. Knot Theory Ramifications 4 (1995), 131-162. MR 96d:57018

[L]
G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257-296. MR 91e:17009

[NZ]
W. D. Nichols and M. B. Zoeller, A Hopf algebra freeness theorem, Amer. J. of Math. 111 (1989), 381-385. MR 90c:16008

[R1]
D. E. Radford, On Kauffman's knot invariants arising from finite-dimensional Hopf algebras, Advances in Hopf algebras (Chicago, 1992), Lecture Notes Pure Appl. Math., vol. 158, Marcel Dekker, New York, 1994, pp. 205-266. MR 96g:57013

[R2]
D. E. Radford, Minimal quasitriangular Hopf algebras, J. of Algebra 157 (1993), 281-315. MR 94c:16052

[RT]
N. Yu. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547-597. MR 92b:57024

[T]
M. Takeuchi, Some Topics on $\mathrm{GL}_q(n)$, J. of Algebra 147(1992), 379-410. MR 93b:17055

[Y]
H. Yamane, A Poincaré-Birkhoff-Witt theorem for quantized universal enveloping algebras of type $A_{N}$, Publ. Res. Inst. Math. Sci. 25 (1989), 503-520. MR 91a:17016

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Additional Information:

Shlomo Gelaki
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720
Email: shlomi@msri.org

Sara Westreich
Affiliation: Interdisciplinary Department of the Social Science, Bar-Ilan University, Ramat-Gan, Israel
Email: swestric@mail.cc.biu.ac.il

DOI: 10.1090/S0002-9947-00-02283-2
PII: S 0002-9947(00)02283-2
Received by editor(s): April 2, 1997
Received by editor(s) in revised form: November 12, 1997
Posted: April 18, 2000
Additional Notes: This research was supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.
Copyright of article: Copyright 2000, American Mathematical Society

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