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Integrals for (dual) quasi-Hopf algebras. Applications. (English) Zbl 1030.16024

This paper studies integrals in quasi-Hopf algebras, the cointegrals for quasi-Hopf algebras introduced by Hausser and Nill, and integrals in dual quasi-Hopf algebras.
Over a field \(k\), a quasi-bialgebra is a four-tuple \((H,\Delta,\varepsilon,\Phi)\) where \(H\) is an associative algebra with unit, \(\Delta\) the comultiplication (which is not coassociative), \(\varepsilon\) the counit and \(\Phi\) the reassociator, an invertible element of \(H\otimes H\otimes H\). A quasi-Hopf algebra is a quasi-bialgebra with antipode \(S\), where \(S\) is an anti-automorphism of \(H\) which satisfies properties similar to those of the antipode of a Hopf algebra. For example, there exist elements \(\alpha,\beta\in H\), such that for all \(h\in H\), \[ \sum S(h_1)\alpha h_2=\varepsilon(h)\alpha\quad\text{and}\quad\sum h_1\beta S(h_2)=\varepsilon(h)\beta. \] It is known that a finite-dimensional quasi-Hopf algebra has a one-dimensional integral space. In this paper, the authors prove that \(\int^H_l\otimes H^*\cong H\) as left \(H\)-modules and their proof does not use the fact that \(S\) is bijective. Thus the bijectivity of \(S\) follows from the other axioms for quasi-Hopf algebras. In fact, a quasi-Hopf algebra \(H\) is finite-dimensional if and only if \(S\) is bijective and \(\int^H_l\neq 0\). The authors also investigate the cointegrals in \(H^*\) introduced by Hausser and Nill for finite-dimensional quasi-Hopf algebras and extend the study of cointegrals to the infinite-dimensional case.
Finally, they study integrals in dual quasi-Hopf algebras and prove that for \(A\) a dual quasi-Hopf algebra, \(A^{*\text{rat}}=0\) if and only if the space of integrals is zero.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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References:

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