Some Classification Results On Finite-dimensional Hopf Algebras

This thesis is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field.It contains two parts:The first part is devoted to classifying pointed Hopf algebras in positive characteristic;The second part is devoted to classifying finite-dimensional Hopf algebras without the dual Chevalley property in characteristic zero.In the first part,by means of the lifting method and the Hochschild cohomology of coalgebras,we obtain a complete classification of pointed Hopf algebras of dimension pg,p2q,4p,2q2 and pqr in positive characteristic p,where p,q,r are distinct prime num-bers.We also classify all pointed Hopf algebras of dimension pq2 that are generated by the first term of the coradical filtration and show that pointed Hopf algebras of dimen-sion pq2 that are not generated by the first term of the coradical filtration must be Hopf extensions of Taft algebras by restricted universal enveloping algebras of dimension p.Then we construct pointed Hopf algebras in positive characteristic whose dimensions can be factorized into at least four prime factors.In particular,we classify all non-connected pointed Hopf algebras of dimension 16 that are generated by the first term of the coradical filtration and non-connected pointed Hopf algebras of dimension 16 whose infinitesimal braidings are one-dimensional.Up to isomorphism,it turns out that there are only finitely many classes of pointed Hopf algebras of dimension pqr,p2q and that there are infinitely many classes of pointed Hopf algebras of dimension pn for any n>2.There are finitely many classes of pointed Hopf algebras of dimension pq2 that are generated by the first term of the coradical filtration.In particular,we obtain some braided Hopf algebras that are not Nichols algebras and infinitely many classes of new non-commutative and non-cocommutative finite-dimensional pointed Hopf algebras.In the second part,by means of the generalized lifting method,we first study the classification of finite-dimensional Hopf algebras over a 4p-dimensional non-pointed basic Hopf algebra Hp,-1 in characteristic zero,where p is a positive integer number.The Hopf algebra Hp,-1 does not have the dual Chevalley property,that is,its coradical is not a subalgebra.In particular,up to isomorphism,Hp,-1 is the only non-pointed basic Hopf algebra of dimension 4p when p is a prime number[29,37].Consider the Drinfeld double D(Hp,-1cop)of Hp,-1cop We determine the projective class ring and the representation type of D(Hp,-1cop).If p = 2,then D(Hp,-1cop)is of tame representation type;otherwise it is of wild type.Then we determine all simple and indecomposable projective objects in Hp,-1Hp,-1 yD by the braided equivalence Hp,-1Hp,-1yD(?)D(Hp,-1cop)M and compute their braidings.Hereafter,we determine all finite-dimensional Nichols algebras in Hp,-1Hp,-1 under the assumption that p is prime.These Nichols algebras are of non-diagonal type.Besides,we construct more examples of Nichols algebras of non-diagonal type in Hp,-1Hp,-1yD and present them by generators and relations when removing the assumption.By computing the liftings of Nichols algebras,we construct families of finite-dimensional Hopf algebras over Hp,-1.In particular,we classify all finite-dimensional Hopf algebras over Hp,-1 whose infinitesimal braidings are indecomposable objects in Hp,-1Hp,-1yD and the diagrams admit non-trivial quadratic relations;we show that all finite-dimensional Hopf algebras over Hp,-1 are basic under the assumption that p>5 is a prprime number;we also classify all finite-dimensional Hopf algebras over H2-,and H3,-1 whose infinitesimal braidings are indecomposable objects.Furthermore,we classify finite-dimensional Hopf algebras over some non-pointed ba-sic Hopf algebras of dimension 16 and 24 whose infinitesimal braidings are indecomposable objects.In particular,we obtain some new Nichols algebras of non-diagonal type and fam-ilies of new finite-dimensional Hopf algebras without the dual Chevalley property.