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Quantum double construction for graded Hopf algebras. (English) Zbl 0801.16037
The main purpose of the paper is the construction of a quasitriangular $$Z_ 2$$-graded Hopf algebra $$(D(A),R)$$ such that for a given $$Z_ 2$$- graded Hopf algebra $$A$$ the Hopf algebra $$D(A)$$ contains $$A$$ and its Hopf dual $$A^ o$$ as Hopf subalgebras. The element $$R$$ is the so called $$R$$- matrix, which is proved to satisfy a graded Yang-Baxter equation (4. Theorem 1). $$D(A)$$ equals the tensor product $$A \otimes A^ o$$ as a linear space, the definition of comultiplication and counit is more or less straightforward, the essential part is the construction of multiplication and antipode. Under the assumption the $$A$$ is proper, i.e. $$A^ o$$ is nontrivial and dense in the algebraic dual space $$A^*$$ of $$A$$, a universal $$R$$-matrix is defined in $$D(A)$$ making it quasitriangular (4. Theorem 3). There is a sketchy application to quantum supergroups. (5. Conclusion).

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 16W50 Graded rings and modules (associative rings and algebras)
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