The Study Of The Freeness Of Nichols Algebras And A Class Of Root Multiplicities

In this thesis we study the structure of some root multiplicities for Nichols algebras of diagonal type.First of all,we focus on the Nichols algebras of diagonal type of rank two.Applying the Poincar?e-Birkhoff-Witt basis of V.Kharchenko,we give a complete expression for the multiplicities of roots of the form8)₁+2₂,for any8)?N₀,over arbitrary fields?the multiplicities of roots8?₁+6)₂,6)?{0,1}are known[44]),where₁,₂is a standard basis of Z².Second,we improve a theorem of Flores de Chela and Green about the determinant of braided symmetrizers.We decide when a Nichols algebra of diagonal type is a free algebra by using an inequality about the number of Lyndon words and an identity about Shuffle maps.Furthermore,under some certain conditions we determine the dimensions of the kernel of the Shuffle maps considered as an operator acting on the free algebra.In 1999,K.V.Kharchenko proved that there exists a Poincar�-Birkhoff-Witt?PBW for short?basis for Nichols algebra of diagonal type,and the generators of the elements of Poincar�-Birkhoff-Witt basis are related to Lyndon words[32].It turns out that the theory of Nichols algebras of diagonal has a break through based on this result.Based on this result,in 2006,I.Heckenberger introduced the of about the generalized root systems and Weyl groupoids of Nichols algebras of diagonal[17],they are generalizations of root systems and Weyl groups of complex semisimple Lie algebras,respectively.It follows that the these generalizations play similar roles as the root system and Weyl group in the theory of complex Lie algebras.Using these tools,I.Heckenberger classified all the finite-dimensional Nichols algebras of of diagonal in terms of Dynkin diagram[18,20,19,22].For the further study of generalized root systems of Nichols algebras of diagonal type is a hard question,but it is necessary.It is well known that for the finite-dimensional Nichols algebras the roots are real roots and multiplicities are one[13,20].However the knowledge about imaginary roots and their multiplicities is little.With our results we make a better understanding of the Nichols algebra theory in this respect.The thesis contains four chapters.In the first chapter,we recall some basic notions and notations,such as Yetter-Drinfel'd modules,braided vector spaces,Nichols algebras,Lyndon words and also some results about Nichols algebras,Lyndon words,etc.For the convenience of expression,we also discuss the notions of root vector candidates and root vectors.In the second chapter,we address the root multiplicities for Nichols algebras of diagonal type of rank two.We concentrate on the root multiplicities of roots of the form8)₁+2₂,8)?N₀,over arbitrary fields.We decide when a root vector candidate is a root vector.In the third chapter,based on an inequality about the number of Lyndon words and an identity about Shuffle maps,we give an explicit condition for when Nichols algebras of diagonal type are free algebras.As an application,we relate the freeness of a particular family of Nichols algebras to the solutions of a quadrat-ic diophantine equation.Under some conditions,we determine the dimensions of the kernel of Shuffle maps.In the last chapter,we study the structure of a class of braided vector space of triangle type.We consider some elements of the algebra of braid group as the automorphism of tensor algebra.We determine the characteristic polynomial of these maps.