Examples of quantum braided groups. (English) Zbl 0832.17020

Hlavatý, Ladislav

Bureš, J. (ed.) et al., Proceedings of the winter school on geometry and physics, Zdíkov, Czech Republic, January 1993. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 37, 93-102 (1994).

Summary: The author gives the defining relations of a new type of bialgebras that generalize both the quantum groups and braided groups as well as the quantum supergroups. The relations of the algebras are determined by a pair of matrices \((R, Z)\) that solve a system of Yang-Baxter-type equations. The matrix coproduct and counit are of standard matrix form, however, the multiplication in the tensor product of the algebra is defined by virtue of the braiding map given by the matrix \(Z\). Besides simple solutions of the system of the Yang-Baxter-type equations that generate either quantum groups or braided groups, we have found several solutions that generate genuine quantum braided groups that by a choice of parameters give quantum or braided groups as a special case.
For the entire collection see [Zbl 0823.00015].

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
16W30	Hopf algebras (associative rings and algebras) (MSC2000)
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory
20F36	Braid groups; Artin groups

Keywords:

defining relations; new type of bialgebras; quantum groups; quantum supergroups; Yang-Baxter-type equations; quantum braided groups