Document Type : Research Paper


Department of Mathematics, Faculty of Science, Azarbaijan shahid Madani university, Tabriz, Iran.


In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(\mathcal{R},\mathcal{H},\mathcal{X})$ consisting of a regular multiplier Hopf algebra $\mathcal{H}$, a left $\mathcal{H}$-comodule algebra $\mathcal{R}$, and a unital left $\mathcal{H}$-module $\mathcal{X}$ which is also a unital algebra. First, we construct a paracyclic module to a triple $(\mathcal{R},\mathcal{H},\mathcal{X})$ and then prove the existence of a cyclic structure associated to this triple.


[1] E. Abe, Hopf algebras, Cambridge University Press (1977).
[2] A. Connes, Noncommutative Geometry, Academic Press (1994).
[3] B. Drabant, A. Van Daele, and Y. Zhang, Actions of multiplier Hopf algebras, Communications in algebra 27 (9) (1999), 4117-4172.
[4] M. Khalkhali, and B. Rangipour, Invariant cyclic homology, $K$-Theory 28 (2) (2003), 183-205.
[5] A. Klimyk and K. Schmudgen, Quantum groups and their representations, Springer-Verlag, Berlin Heidelberg,1997.
[6] J.L. Loday, Cyclic Homology, Springer-Verlag, (1992).
[7] A. Van Daele, An algebraic framework for group duality, Adv. in math. 140 (1998), 323-366.
[8] A. Van Daele, Multiplier Hopf algebras, Trans. Amer. Math. Soc., 342 (1994), 917-932.
[9] A. Van Daele and Y.H. Zhang, Galois Theory for Multiplier Hopf Algebras with Integrals, Algebras and representation theory, 2 (1999), 83-106.