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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.

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