Cocycle deformations and Galois objects for some cosemisimple Hopf algebras of finite dimension (Q2708915)

The author studies 4 families of nonisomorphic cosemisimple Hopf algebras of dimension \(2n\) over a field \(k\) of characteristic different from 2, denoted \(\widehat{\mathcal D}_{2n}\), \(\widehat{\mathcal T}_{4m}\), \({\mathcal A}_{4m}\), \({\mathcal B}_{4m}\) where in the last 3 cases, \(n=2m\) is even. The first two classes are commutative Hopf algebras; if \(k\) has a primitive \(n\)-th root of 1 then \(\widehat{\mathcal D}_{2n}\cong k^G\) for \(G=D_{2n}\), the dihedral group and \(\widehat{\mathcal T}_{4m}\cong k^G\) for \(G=T_{4m}\) the dicyclic group. The Hopf algebras \({\mathcal A}_{4m}\), \(m\geq 3\), and \({\mathcal B}_{4m}\), \(m\geq 2\), are neither commutative nor cocommutative. If \(k\) contains a square root of \(-1\), then \({\mathcal B}_8\) is isomorphic to the unique semisimple Hopf algebra of dimension 8 which is neither commutative nor cocommutative.NEWLINENEWLINENEWLINEA deformation of a Hopf algebra \(H\) by an invertible cocycle \(\sigma\) is the Hopf algebra \(H^\sigma\) which is the coalgebra \(H\) with product twisted by \(\sigma\), i.e. NEWLINE\[NEWLINEx\cdot y=\sum\sigma(x_1,y_1)x_2y_2\sigma^{-1}(x_3,y_3).NEWLINE\]NEWLINE If \(L=H^\sigma\), then there is a monoidal equivalence \(H\text{-Comod}\approx L\text{-Comod}\) and the converse holds if \(H\) is finite-dimensional. Also there is a 1-1 correspondence between \((L,H)\)-biGalois objects and \(k\)-linear monoidal equivalences \(H\text{-Comod}\to L\text{-Comod}\) since a \(k\)-linear monoidal equivalence \(H\text{-Comod}\approx H'\text{-Comod}\) is given by the cotensor product \(R\square_H\) where \(R\) is a uniquely chosen \((H,H')\)-biGalois object.NEWLINENEWLINENEWLINEThe main theorem of this paper is:NEWLINENEWLINENEWLINETheorem. Suppose \((k^\times)^{2n}=k^\times\). (1) Any of \(\widehat{\mathcal D}_8=k^{D_8}\), \(\widehat{\mathcal D}_{2n}\) (\(n\) odd), \(\widehat{\mathcal T}_{4m}\), \({\mathcal B}_{4m}\) has no nontrivial deformation. (2) If \(m\geq 3\), \(\widehat{\mathcal D}_{4m}\) and \({\mathcal A}_{4m}\) are deformations of each other and these are not deformed from any other Hopf algebras.NEWLINENEWLINENEWLINEThe last theorem of the paper describes the Galois objects for these Hopf algebras.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00038].