Abstract
We develop a theory of bornological quantum hypergroups, aiming to extend the theory of algebraic quantum hypergroups in the sense of Delvaux and Van Daele to the framework of bornological vector spaces. It is very similar to the theory of bornological quantum groups established by Voigt, except that the coproduct is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We study the Fourier transform and develop Pontryagin duality theory for a bornological quantum hypergroup. As an application, we prove a formula relating the fourth power of the antipode with the modular functions of a bornological quantum hypergroup and its dual.
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References
Caenepeel S., Militaru G., Zhu S.: Frobenius and separable functors generalized module categories and nonlinear equations. Lecture Notes in Mathematics 1787. Springer Verlag, Berlin (2002)
Delvaux, L., Van Daele, A.: Algebraic quantum hypergroups, [math. RA] 12 April 2007, arXiv: math(0606466v3)
Hogbe-Nlend H.: Bornologies and functionals analysis, Notas de Matemática 62. North-Holland, Amsterdam (1977)
Meyer R.: Smooth group representations on bornological vector spaces. Bull. Sci. Math. 128(2), 127–166 (2004)
Van Daele, A.: The Fourier Transform in Quantum Group Theory. New Techniques in Hopf Algebras and Graded Ring Theory, pp. 187–196, K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels (2007)
Van Daele A.: Multiplier Hopf algebras. Trans. Am. Math. Soc. 342(2), 917–932 (1994)
Van Daele A.: An algebraic framework for group duality. Adv. Math. 140, 323–366 (1998)
Van Daele A.: Tools for working with multiplier Hopf algebras. Arab. J. Sci. Eng. 33(2C), 505–527 (2008)
Voigt C.: Bornological quantum groups. Pacific J. Math. 235(1), 93–135 (2008)
Voigt C.: Equivariant periodic cyclic homology. J. Inst. Math. Jussieu 6(4), 689–763 (2007)
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Van Daele, A., Wang, S. Pontryagin duality for bornological quantum hypergroups. manuscripta math. 131, 247–263 (2010). https://doi.org/10.1007/s00229-009-0318-8
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DOI: https://doi.org/10.1007/s00229-009-0318-8