Hopf *-algebra structures on H(1, q)

Abstract

We study the Hopf *-algebra structures on the Hopf algebra H(1, q) over ℂ. It is shown that H(1, q) is a Hopf *-algebra if and only if |q| = 1 or q is a real number. Then the Hopf *-algebra structures on H(1, q) are classified up to the equivalence of Hopf *-algebra structures.

Keywords

*-Structure / antilinear map / Hopf *-algebra

References

[1] Chen H X. A class of noncommutative and noncocommutative Hopf algebras—the quantum version. Comm Algebra, 1999, 27: 5011−5023

[2] Chen H X. Irreducible representations of a class of quantum doubles. J Algebra, 2000, 225: 391−409

[3] Chen H X. Finite-dimensional representations of a quantum double. J Algebra, 2002, 251: 751−789

[4] Chen H X. Representations of a class of Drinfeld’s doubles. Comm Algebra, 2005, 33: 2809−2825

[5] Kassel C. Quantum Groups. New York: Springer-Verlag, 1995

[6] Majid S. Foundations of Quantum Group Theory. Cambridge: Cambridge Univ Press, 1995

[7] Masuda T, Mimachi K, Nagakami Y, Noumi M, Saburi Y, Ueno K. Unitary representations of the quantum group SU_q(1.1): Structure of the dual space of U_q(sl(2)). Lett Math Phys, 1990, 19: 187−194

[8] Montgomery S. Hopf Algebras and Their Actions on Rings. CBMS Reg Conf Ser Math, No 82. Providence: Amer Math Soc, 1993

[9] Sweedler M E. Hopf Algebras. New York: Benjamin, 1969

[10] Wang Z, Chen H X. Generic modules over a class of Drinfeld’s quantum doubles. Comm Algebra, 2008, 36: 3730−3749

[11] Woronowicz S L. Compact matrix pseudo-groups. Comm Math Phys, 1987, 111: 613−665

[12] Woronowicz S L. Twisted SU(N) group. An example of non-commutative differential calculus. Publ Res Inst Math Sci, 1987, 23: 117−181

[13] Woronowicz S L. Tannaka-Krein duality for compact matrix pseudo-groups. Twisted SU(N) groups. Invent Math, 1988, 93: 35−76

[14] Zhang Y, Wu F, Liu L, Chen H X. Grothendieck groups of a class of quantum doubles. Algebra Colloq, 2008, 15: 431−448

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Higher Education Press and Springer-Verlag Berlin Heidelberg