Group-Graded By-Product Construction and Group Double Centralizer Properties

Abstract

For a group π with unit e, we introduce and study the notion of a π-graded Hopf algebra. Then we introduce and construct a new braided monoidal category HHeYDπ over a π-graded Hopf algebra H. We introduce the notion of a π-double centralizer property and investigate this property by studying a braided π-graded Hopf algebra U(gln(V))⋉πH, where V is an n-dimensional vector space in HHeYDπ and U(gln(V)) is the braided universal enveloping algebra of gln(V) which is not the usual Hopf algebra. Finally, some examples and special cases are given.

Keywords

• group-graded Hopf algebras
• symmetric braided categories
• group Yetter–Drinfeld categories
• group double centralizer properties