Hopf algebras of types and which give rise to certain invariants of knots, links and 3-manifolds

Authors:
Shlomo Gelaki and Sara Westreich

Journal:
Trans. Amer. Math. Soc. **352** (2000), 3821-3836

MSC (2000):
Primary 16W30

Published electronically:
April 18, 2000

MathSciNet review:
1491865

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

In this paper we determine when Lusztig's 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.

**[D1]**V. G. Drinfel′d,*Almost cocommutative Hopf algebras*, Algebra i Analiz**1**(1989), no. 2, 30–46 (Russian); English transl., Leningrad Math. J.**1**(1990), no. 2, 321–342. MR**1025154****[D2]**V. G. Drinfel′d,*Hopf algebras and the quantum Yang-Baxter equation*, Dokl. Akad. Nauk SSSR**283**(1985), no. 5, 1060–1064 (Russian). MR**802128****[GW]**Shlomo Gelaki and Sara Westreich,*On the quasitriangularity of 𝑈_{𝑞}(𝑠𝑙_{𝑛})’*, J. London Math. Soc. (2)**57**(1998), no. 1, 105–125. MR**1624793**, 10.1112/S0024610798005705**[H]**Mark Hennings,*Invariants of links and 3-manifolds obtained from Hopf algebras*, J. London Math. Soc. (2)**54**(1996), no. 3, 594–624. MR**1413901**, 10.1112/jlms/54.3.594**[J]**M. Jimbo, Quantum groups and the Yang-Baxter equation, to appear.**[K]**Louis H. Kauffman,*Gauss codes, quantum groups and ribbon Hopf algebras*, Rev. Math. Phys.**5**(1993), no. 4, 735–773. MR**1253734**, 10.1142/S0129055X93000231**[KR1]**Louis H. Kauffman and David E. Radford,*A necessary and sufficient condition for a finite-dimensional Drinfel′d double to be a ribbon Hopf algebra*, J. Algebra**159**(1993), no. 1, 98–114. MR**1231205**, 10.1006/jabr.1993.1148**[KR2]**Louis H. Kauffman and David E. Radford,*Invariants of 3-manifolds derived from finite-dimensional Hopf algebras*, J. Knot Theory Ramifications**4**(1995), no. 1, 131–162. MR**1321293**, 10.1142/S0218216595000077**[L]**George Lusztig,*Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra*, J. Amer. Math. Soc.**3**(1990), no. 1, 257–296. MR**1013053**, 10.1090/S0894-0347-1990-1013053-9**[NZ]**Warren D. Nichols and M. Bettina Zoeller,*A Hopf algebra freeness theorem*, Amer. J. Math.**111**(1989), no. 2, 381–385. MR**987762**, 10.2307/2374514**[R1]**David E. Radford,*On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras*, Advances in Hopf algebras (Chicago, IL, 1992) Lecture Notes in Pure and Appl. Math., vol. 158, Dekker, New York, 1994, pp. 205–266. MR**1289427****[R2]**David E. Radford,*Minimal quasitriangular Hopf algebras*, J. Algebra**157**(1993), no. 2, 285–315. MR**1220770**, 10.1006/jabr.1993.1102**[RT]**N. Reshetikhin and V. G. Turaev,*Invariants of 3-manifolds via link polynomials and quantum groups*, Invent. Math.**103**(1991), no. 3, 547–597. MR**1091619**, 10.1007/BF01239527**[T]**Mitsuhiro Takeuchi,*Some topics on 𝐺𝐿_{𝑞}(𝑛)*, J. Algebra**147**(1992), no. 2, 379–410. MR**1161300**, 10.1016/0021-8693(92)90212-5**[Y]**Hiroyuki Yamane,*A Poincaré-Birkhoff-Witt theorem for quantized universal enveloping algebras of type 𝐴_{𝑁}*, Publ. Res. Inst. Math. Sci.**25**(1989), no. 3, 503–520. MR**1018513**, 10.2977/prims/1195173355

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
16W30

Retrieve articles in all journals with MSC (2000): 16W30

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:
http://dx.doi.org/10.1090/S0002-9947-00-02283-2

Received by editor(s):
April 2, 1997

Received by editor(s) in revised form:
November 12, 1997

Published electronically:
April 18, 2000

Additional Notes:
This research was supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.

Article copyright:
© Copyright 2000
American Mathematical Society