Some Aspects Of Cyclic Homology And Quantum Quasi-Shuffle Algebras

In this thesis, we study three topics on cyclic homology theory:cyclic homology of strong smash product algebras, Hopf-cyclic homology of Bichon's algebra, and a "nat-ural" graded Hopf algebra and its graded Hopf-cyclic cohomology. Also we study a relatively independent topic:quantum quasi shuffle algebras.This work is divided into four chapters. Each topic is discussed in one chapter.Calculating cyclic homology of some crossed product algebras is an important and difficult problem in cyclic homology theory. In Chapter 2, we define a more general class of so-called strong smash product algebras, denoted by A#RB, for given two algebras A and B with an invertible morphism R mapping from B(?) A to A(?) B. With the dia-grammatical presentation deriving from the braided tensor category and knot theory, we construct a cylindrical module A(?)B whose diagonal cyclic module△,(A(?)B) is graphi-cally proved to be isomorphic to C.(A#RB) the cyclic module of the algebra. Then using the generalized Eilenberg-Zilber Theorem, we establish a spectral sequence to converge to the cyclic homology of A#RB.In particular, we prove that when the antipodes of the Hopf algebras concerned are invertible, the classical crossed product, Takeuchi's smash product, Majid's double crossproduct of Hopf algebras are all strong smash product algebras. Especially, many al-gebraic objects being of Drinfeld's quantum double structure, can be realized as Majid's double crossproduct of Hopf algebras. Therefore, the notion of strong smash product algebras does cover a wild range of the recent interesting examples, for instance, the two-parameter or multiparameter quantum groups, and the pointed Hopf algebras arising from the classification of Nichols algebras of diagonal type, etc. So, the cyclic homology the-ory of strong smash product algebras can be used in a much more general framework. Moreover, since the two subalgebras with interactions on each other play a balanced role in the strong smash product algebra, it allows us to take advantage of the good homolog-ical property of either of them in many cases to do computation effectively. At the end of this chapter, a couple of concrete examples, which can not be computed or is more complicated to be computed in the framework of [22] or [1], are given to illuminate our results.In Chapter 3, we compute the Hopf-cyclic homology of Bichon's algbraβN (where N is a prime) with non trivial modular pairs in involution. We give some new q-identities on Gaussian polynomial which are very useful in our computation of the Hopf-cyclic homology. Then we construct a "small" free resolution of K as BN-modules which can be used to compute the Hochschild homology of Bichon's algebra with coefficients in K more easily. The importance of Bichon's algebra lies in the fact that the category of BN-comodule algebras is monoidally equivalent to the category of N-differential graded alge-bras, which are related to N-complexes studied by Kassel, Wambst, Berger and Dubois-Violette, etc. In Chapter 4, we find a special graded Hopf algebra and prove that the category of differential graded algebras is monoidally equivalent to the category of left graded comodule algebras over that graded Hopf algebra. After calculating the graded Hopf-cyclic cohomology of that graded Hopf algebra, we construct cyclic cocycles on any graded differential algebra with closed graded trace by means of a characteristic ho-momorphism.In Chapter 5, we establish some properties of quantum quasi-shuffle algebras. They include the necessary and sufficient condition for the construction of the quantum quasi-shuffle product, the universal property, and the commutativity condition. As an applica-tion, we use the quantum quasi-shuffle product to construct a linear basis of T(V), for a special kind of Yang-Baxter algebras (V, m, a).