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...

The cross coproduct braided group Aut(C)◮⊳Aut(C) and higher braided group ‘spin chains’. The example of the braided...

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of...

We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors.

Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac–Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwist...

We compute the braided groups and braided matrices B(R) for the solution R of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the Heisenberg universal enveloping algebra U q (h), and use this result to derive an action of U q (h) on the braided groups. We then demonstrate the various covaria...

We introduce braided Lie bialgebras as the infinitesimal version of braided groups. They are Lie algebras and Lie coalgebras with the coboundary of the Lie cobracket an infinitesimal braiding. We provide theorems of transmutation, Lie biproduct, bosonisation and double-bosonisation relating braided Lie bialgebras to usual Lie bialgebras. Among the results, the kernel of any split projection of ...

∗-structures on quantum and braided spaces of the type defined via an R-matrix are studied. These include q-Minkowski and q-Euclidean spaces as additive braided groups. The duality between the ∗-braided groups of vectors and covectors is proved and some first applications to braided geometry are made.

Let A be a Hopf algebra in a braided category C. Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category DY (C)AA of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group (A,A,R) the correspo...

We find a new braided Hopf structure for the algebra satisfied by the entries of the braided matrix BSLq(2). A new nonbraided algebra whose coalgebra structure is the same as the braided one is found to be a two parameter deformed algebra. It is found that this algebra is not a comodule algebra under adjoint coaction. However, it is shown that for a certain value of one of the deformation param...