The Drinfel'd double of multiplier Hopf algebras.
Let \(\langle A,B\rangle\) be a pairing of two regular multiplier Hopf algebras \(A\) and \(B\) to \(\mathbb{C}\). This basically means that the comultiplication in each is dual to the multiplication in the other, but there are some other technical conditions [see \textit{B. Drabant} and \textit{A. Van Daele}, Algebr. Represent. Theory 4, No. 2, 109-132 (2001; Zbl 0993.16024)]. The Drinfeld double \(D\) of this pairing was constructed by Drabant and Van Daele [op. cit.] and by \textit{L. Delvaux} [J. Algebra 269, No. 1, 285-316 (2003; Zbl 1036.16030)], in a manner similar to the construction of the Drinfeld double for Hopf algebras. In the paper under review, the Drinfeld double \(D\) of \(\langle A,B\rangle\) is realized as an algebra of operators on the vector space \(B\otimes A\). If \(\langle A,B\rangle\) is a multiplier Hopf *-algebra pairing, then \(D\) is also a multiplier Hopf *-algebra. A non-zero linear functional \(\varphi\) in \(A^*\) is called a left integral if \((I\otimes\varphi)\Delta(a)=\varphi(a)1\) for \(a\) in \(A\). If \(A\) is a multiplier Hopf *-algebra, \(\varphi\) is called positive if \(\varphi(aa^*)\geq 0\) for \(a\) in \(A\). The authors show that if \(A\) and \(B\) are multiplier Hopf *-algebras with positive left integrals, then their Drinfeld double \(D\) also has a positive left integral. This was announced in the two earlier papers cited above, and a somewhat involved proof is given in the paper under review.
scientific article; zbMATH DE number 2057673
scientific article