We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnar’s smash coproduct Hopf algebra. We find connections between the exponent and Frobenius-Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius-Schur indicators of the original Hopf algebra A. We study the category of modules of ...

We consider the algebra isomorphism found by Frenkel and Ding between the RLL and the Drinfeld realizations of Uq(ĝl(2)). After we note that this is not a Hopf algebra isomorphism, we prove that there is a unique Hopf algebra structure for the Drinfeld realization so that this isomorphism becomes a Hopf algebra isomorphism. Though more complicated, this Hopf algebra structure is also closed, ju...

A generalized oscillator algebra is proposed and the braided Hopf algebra structure for this generalized oscillator is investigated. Using the solutions for the braided Hopf algebra structure, two types of braided Fibonacci oscillators are introduced. This leads to two types of braided Biedenharn-Macfarlane oscillators as special cases of the Fibonacci oscillators. We also find the braided Hopf...

For G a finite abelian group, we study the properties of general equivalence relations on Gn = Gn Sn , the wreath product of G with the symmetric group Sn , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of kGn as well as graded connected Hopf subalgebras of ⊕ n≥o kGn . In particular we construct a G-colou...

A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ : H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasi-symmetric functions; this explains the ubiquity of quasi-symmetric functions as generating functions in combinatorics. We illustrate this...

We find a formula to compute the number of the generators, which generate the n-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight n. Applying Hopf algebra of rooted trees, we show that the analogue of Andruskiewitsch and Schneider’s Conjecture is not true. The Hopf algebra of rooted trees and the enveloping algebra of the Lie algeb...