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Multiplication alteration by two-cocycles. The quantum version. (English) Zbl 0821.16038
Let \(A\) be a bialgebra over a field \(k\), \(\sigma\) a convolution- invertible 2 cocycle from \(A\otimes A\) to \(k\). Define a new multiplication \(*\) on \(A\) by \(x*y= \sum \sigma(x_ 1, y_ 1) x_ 2 y_ 2 \sigma^{-1} (x_ 3, y_ 3)\), where \(\sum x_ 1\otimes x_ 2 \otimes x_ 3\) is the double diagonalization of \(x\). With this new multiplication, and the original comultiplication, \(A\) becomes a new bialgebra, denoted \(A^ \sigma\). This construction was given by the first author [Commun. Algebra 21, 1731–1749 (1993; Zbl 0779.16015)].
Here the authors first show that the Drinfeld double \(D(H)\) of a finite-dimensional Hopf algebra \(H\) is of the form \(A^ \sigma\), where \(A= (H^* )^{\text{coop}} \otimes H\), and \(\sigma (x\otimes a, y\otimes b)= x(1) y(a) \varepsilon (b)\), \(\varepsilon\) the augmentation of \(H\). This is generalized to skew-paired bialgebras giving a natural extension of Drinfeld’s \(D(H)\). Applications to braided bialgebras are given, i.e., a bialgebra skew-paired with itself. For example, the quantum matrix bialgebra \(M_ q (n)\) and the quantum enveloping algebra \(U_ q(\text{GL}_ n (q))\) when \(q\) is not a root of unity, can be realized in this way.

16T10 Bialgebras
16T20 Ring-theoretic aspects of quantum groups
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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