Braided cofree Hopf algebras and quantum multi-brace algebras

Generalized Matsumoto–Tits sections and quantum quasi-shuffle algebras

“…By definition ( [18], [17]), the quantum quasi-shuffle algebra Q σ,m (V ) is the subalgebra of T σ,m (V ) generated by V . Another direct definition of the total symmetrization operator is given by Jian [16] by generalizing the construction of Guo and Keigher ([13]), compared to their results, our approach makes the combinatorial aspects of the quantum quasi-shuffle product (quasi-shuffle product, quantum shuffle product) more transparent and can be applied not only to the extremal cases (i.e., |I| = 1 and |I| = n − 1) as in [16] but also to an arbitrary descent.…”

“…• Free Rota-Baxter algebras ( [3], [13]); • Positive parts of quantum groups ( [25], [26], [12]); the entire quantum group ( [10]); • Multiple zeta values ( [14]) and their q-deformed versions ( [15]); • Quasi-symmetric functions and their q-deformed versions ( [27]); • Free dendriform and tridendriform algebras ( [20]); • Hopf algebra of rooted trees ( [5]); • (Conjecturally) Fomin-Kirillov algebras ( [11], [22]). These products are originally defined using either universal properties [17] or by inductive formulae [18]. Its combinatorial nature is first studied by Hoffman [14], Guo-Keigher [13] and Ebrahimi-Fard-Guo [6] in the classical case (quasi-shuffle products) and henceforth generalized by Jian [16] to the quantum case using the notion of mixable shuffles [6].…”

“…Concretely, the construction of the quantum quasi-shuffle product associates to a braided algebra (V, σ, m) a multiplicative structure (the quantum quasi-shuffle product) on the tensor space T (V ) involving the multiplication m via the universal property ( [17], [18]). The output is denoted by T σ,m (V ) and its sub-algebra Q m,σ (V ) generated by V is called the quantum quasi-shuffle algebra.…”

Construction of Rota-Baxter algebras via Hopf module algebras

“…It is shown, in [18], that any modulealgebra A over H is moreover an algebra in H H YD. Then Rota-Baxter operators P R and P L on A#H are given respectively by P R (a#h) = a#ε(h)1 H ,…”

Cofree Hopf algebras on Hopf bimodule algebras

“…Thanks to this isomorphism, we can concentrate our study on quantum quasi-shuffle algebras, which are the quantization of the classical quasishuffle algebras introduced by Newman and Radford [15], Hoffmann ( [7]), Guo and Keigher ( [6]) independently. They are constructed as a special case of quantum multibrace algebras ( [11]) and have some interesting properties and applications to other mathematical objects (cf. [9], [10], [12]).…”