Generalized Matsumoto–Tits sections and quantum quasi-shuffle algebras

On quantum shuffle and quantum affine algebras

A construction of the quantum affine algebra Uq(ĝ) is given in two steps. We explain how to obtain the algebra from its positive Borel subalgebra Uq(b +), using a construction similar to Drinfeld’s quantum double. Then we show how the positive Borel subalgebra can be constructed with quantum shuffles.

Generalized quantum current algebras

Two general families of new quantum deformed current algebras are proposed and identified both as infinite Hopf family of algebras, a structure which enable one to define “tensor products” of these algebras. The standard quantum affine algebras turn out to be a very special case of both algebra families, in which case the infinite Hopf family structure degenerates into standard Hopf algebras. T...

Baxter Algebras and Shuffle Products ∗

Quasi-Shuffle Products

Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication ∗ on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product III . We extend this commutative algebra structure to a Hopf algebra (A, ∗,1); in the case where A is the s...

Quantum Double for Quasi-hopf Algebras

We introduce a quantum double quasitriangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D(G) associated to a finite group G and group 3-cocycle φ. We also discuss D(Ug) as...