Some Results On Finite Dimensional Nichols Algebras

Nichols algebras play a central role in the theory of (pointed) Hopf alge-bras. It comes from the Lifting Method by Andruskiewitsch and Schneider toclassify fnite dimensional pointed Hopf algebras. Any braided vector space has acanonical Nichols algebra. The easiest braidings are those of diagonal type. Thefnite-dimensional Nichols algebra of diagonal type with braided vector spaces werealmost classifed by Heckenberger.We prove the main results in this paper as fol-lows:(i) Nichols algebra B(V) is fnite-dimensional if and only if Nichols braidedLie algebra L(V) is fnite-dimensional.(ii) All fnite dimensional Nichols algebraswith diagonal type of connected fnite dimensional Yetter-Drinfeld(YD in short)modules over fnite cyclic group Znare found. It is proved that Nichols algebra ofconnected YD module V over Znwith dim V>3is infnite dimensional.(iii) Weprove that except in several cases Nichols algebras of irreducible YD modules overclassical Weyl groups Zn2Snare infnite dimensional and bi-one arrow Nicholsalgebras and B(Os, ρ) are the same up to isomorphisms.In Chapter2, based on Heckenberger research work we compare the dimensionof a Nichols algebra of diagonal type with the dimension of the correspondingNichols braided Lie algebra and related structures. In section1we recall someresults on Nichols algebras and fx the notation. In section2we show that L (V)is infnite dimensional if D is infnite. In section3we prove that B(V) is fnite-dimensional if and only if L(V) is fnite-dimensional. In section4we present thesufcient conditions for B(V)=F⊕L(V). In section5we give the sufcientconditions for Nichols braided Lie algebra L(V) to be a homomorphic image of abraided Lie algebra generated by V with defning relations.In Chapter3, we study diagonal braidings and their Nichols algebras comingfrom YD modules over fnite cyclic groups. In sections1and2we fnd all fnitedimensional Nichols algebras with diagonal type of connected2-dimensional and3-dimensional Zn-YD modules, respectively. In section3we prove that Nicholsalgebra with diagonal type of connected Zn-YD module V with dim V>3isinfnite dimensional.In Chapter4, frst we apply juxtapositions to decide if Nichols algebras associ-ated to the irreducible YD modules over classic Weyl groups are fnite dimensionalor not. In section2we prove that except in several cases Nichols algebras of irre-ducible YD modules over classical Weyl groups Zn2Snare infnite dimensional. In section3we prove that except in several cases conjugacy classes of classicalWeyl groups are of type D. In section4it is shown that bi-one arrow Nicholsalgebras and B(Os, ρ) are the same up to isomorphisms. In section5we givethe relationship between FK’ Conjecture and Nichols algebra B(O(1,2), sgn) oftransposition over symmetry group by means of quiver Hopf algebras. That is, ifdim B(O(1,2), sgn)=∞, then so is dim En. And we proposed a more generalproblem under the basis of FK conjecture.