Finite Dimensional Rank 2 Nichols Algebras Of Diagonal Type And Braided Lie Algebra

It is well known that,for the finite dimensional rank 2 Nichols algebras of diagonal type,describing the generators and relations of them and computing their dimensions have important significance.This article is based on the re-search of Heckenberger's two-dimensional Nichols algebras of diagonal type,he classified the type of the finite dimensional rank 2 Nichols algebras of diagonal type(22),further,all relations of them are elements of the ideal of V(?) generated by the setThis article calculated the specific formula and obtained specific results:(ⅰ).λ(c)=λ(sa)=λ(a)+χ(a,aR)-1(s+1)pa-χ(aR,a)(s+1)pa-1,.where c:=sa:=aR,L…L,the number of L is s,and lR(sa)=s+1.(ⅱ).λ(b)=λ(a)+χ(aL,a)-1(s+1)pa-χ(a,aL)(s+1)pa-1,where b=as:= aL,R…R,the number of R is s,and lL(as)=s+1(ⅲ).λ(a)=q21-2(lL(as))q11-1-q12(lL(as))q11 where a=as:=aR,R…R,aR=R and lL(as)=s+1.(ⅴ).λ(a)=q21-1(lR(sa))q22-1-q12(lR(sa))q22 where a=sa:=aL,L…L,aL=L, and lR(sa)=s+1.It provides the basis for further study on the relations and generators.Color algebras,m-braid algebras have widely applications in the exchange of algebraic geometry.Because of the complexity of the braid,the study on the m-braid algebras stays only on the basic concept of its structure,there is no depth study on their structure.In this paper,by introduction the braiding c,we give the general Jacobi identity of the braided Lie algebraThis article using the combination of mathematics and the concept of Lyndon words draw the enveloping algebra of m-braided algebras(PBW).(U,φ)is the enveloping algebra of L.