Group-Graded By-Product Construction and Group Double Centralizer Properties - Senlin Zhang; Shuanhong Wang (Mathematics, 2022)
For a group <semantics>π</semantics> with unit e, we introduce and study the notion of a <semantics>π</semantics>-graded Hopf algebra. Then we introduce and construct a new braided monoidal category <semantics><msubsup>HHe</msubsup>YDπ</semantics> over a <semantics>π</semantics>-graded Hopf algebra H. We introduce the notion of a <semantics>π</semantics>-double centralizer property and investigate this property by studying a braided <semantics>π</semantics>-graded Hopf algebra <semantics>U(gln(V))⋉πH</semantics>, where V is an n-dimensional vector space in <semantics><msubsup>HHe</msubsup>YDπ</semantics> and <semantics>U(gln(V))</semantics> is the braided universal enveloping algebra of <semantics>gln(V)</semantics> which is not the usual Hopf algebra. Finally, some examples and special cases are given.