Abstract
Finite dimensional ribbon Hopf (super) algebras play an important role in constructing invariants of 3-manifolds. In the present paper, the authors give a necessary and sufficient condition for the Drinfel’d double of a finite dimensional Hopf superalgebra to have a ribbon element. The criterion can be seen as a generalization of Kauffman and Radford’s result in the non-super situation to the ℤ2-graded situation, however, the derivation of the result in the ℤ2-graded case will be much more complicated.
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References
Chen, Q. and Wang, D., Constructing quasitriangular Hopf algebras, Comm. Alg., 43(4), 2015, 1698–1722.
Wang, D., Zhang, J. J. and Zhuang, G. B., Primitive cohomology of Hopf algebras, J. Algebra, 464, 2016, 36–96.
Gould, M. D., Zhang, R. B. and Bracken, A. J., Quantum double construction for graded Hopf algebras, Bull. Austral. Math. Soc., 47(3), 1993, 353–375.
Reshetikhin, N. and Turaev, V. G., Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math., 103(1), 1991, 547–597.
Kauffman, L. H. and Radford, D. E., A necessary and sufficient condition for a finite-dimensional Drinfel’d double to be a ribbon Hopf algebra, J. Algebra, 159(1), 1993, 98–114.
Benkart, G. and Witherspoon, S., Restricted two-parameter quantum groups, Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry, Fields Inst. Commun., 40, Amer. Math. Soc., Providence, RI, 2004, 293–318.
Hu, N. and Wang, X., Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of type B, J. Geom. and Phys., 60, 2010, 430–453.
Hu, N. and Wang, X., Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of type G 2, Pacific J. Math., 241(2), 2009, 243–273.
Dong, H. and Huang, H., Hopf Superquivers, Chin. Ann. Math. Ser. B., 32(2), 2011, 253–264.
Blumen, S., Quantum superalgebra at roots of unity and topological invariants of three-manifolds, Univ of Sydney, Australia, 2005.
Fischman, D., Montgomery, S. and Schneider, H.-J., Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math. Soc., 349(12), 1997, 4857–4895.
Radford, D. E., Minimal quasitriangular Hopf algebras, J. Algebra, 157(2), 1993, 285–315.
Hennings, M., Invariants of links and 3-manifolds obtained from Hopf algebras, J. London Math. Soc., 54, 1996, 594–624.
Liu, K., A family of new universal R-matrix, Proc. Amer. Math. Soc., 125(4), 1997, 987–999.
Panaite, F., Ribbon and charmed elements for quasitriangular Hopf algebras, Comm. Alg., 25(3), 1997, 973–977.
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This work was supported by the National Natural Science Foundation of China (Nos. 11701019, 11671024) and the Natural Science Foundation of Beijing (No. 1162002).
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Chen, J., Yang, S. Ribbon Hopf Superalgebras and Drinfel’d Double. Chin. Ann. Math. Ser. B 39, 1047–1064 (2018). https://doi.org/10.1007/s11401-018-0113-5
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DOI: https://doi.org/10.1007/s11401-018-0113-5