秩为2的对角型Nichol代数与辫子李代数
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摘要
众所周知,关于秩为2的对角型Nichol代数,给出其生成子及关系,并计算它们的维数有着重要的意义。本文正是基于Heckenbergerl的秩为2的对角型Nichol代数的研究,他分类了秩为2的有限维Nichol代数的种类(22类),得出其所有的关系由集生成的V(?)的理想的元素。
而本文则计算出了具体生成子公式：
(ⅰ).λ(c)=λ(sa)=λ(a)+χ(a,aR)-1(s+1)pa-χ(aR,a)(s+1)pa-1.其中c=sa=aR,L…L,的个数为s,且lR(sa)=s+1.
(ⅱ).λ(b)=λ(a)+χ(aL,a)-1(s+1)pa-χ(a,aL)(s+1)pa-1其中b=as:=aL,R…R,R的个数为s个,且lL(as)=s+1
(ⅲ).λ(a)=q21-2(lL(as))q11-1-q12(lL(as))q11其中a=as:=aR,R…R,aR=R且lL(as)=s+1.
(ⅴ).λ(a)=q21-1(lR(sa))q22-1-q12(lR(sa))q22其中a=sa:=aL,L…L,aL=L,且lR(sa)=s+1
计算出其相关数据,得出具体结果,对进一步研究其生成子与关系提供了基础。
色李代数,m-辫子李代数在非交换代数几何中有着广泛的应用。由于辫子的复杂性,对m-辫子李代数的研究只停留在基本的概念上,对它的结构并未作深入研究。本文通过引入辫子,给出了一般的辫子李代数的Jacobi等式并利用组合数学中Lydon字的概念给出了m-辫子李代数的泛包络代数。
(PBW).(U,φ)是辫子李代数L的包络代数。
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 set
This 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 algebra
This 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.
而本文则计算出了具体生成子公式：
(ⅰ).λ(c)=λ(sa)=λ(a)+χ(a,aR)-1(s+1)pa-χ(aR,a)(s+1)pa-1.其中c=sa=aR,L…L,的个数为s,且lR(sa)=s+1.
(ⅱ).λ(b)=λ(a)+χ(aL,a)-1(s+1)pa-χ(a,aL)(s+1)pa-1其中b=as:=aL,R…R,R的个数为s个,且lL(as)=s+1
(ⅲ).λ(a)=q21-2(lL(as))q11-1-q12(lL(as))q11其中a=as:=aR,R…R,aR=R且lL(as)=s+1.
(ⅴ).λ(a)=q21-1(lR(sa))q22-1-q12(lR(sa))q22其中a=sa:=aL,L…L,aL=L,且lR(sa)=s+1
计算出其相关数据,得出具体结果,对进一步研究其生成子与关系提供了基础。
色李代数,m-辫子李代数在非交换代数几何中有着广泛的应用。由于辫子的复杂性,对m-辫子李代数的研究只停留在基本的概念上,对它的结构并未作深入研究。本文通过引入辫子,给出了一般的辫子李代数的Jacobi等式并利用组合数学中Lydon字的概念给出了m-辫子李代数的泛包络代数。
(PBW).(U,φ)是辫子李代数L的包络代数。
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 set
This 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 algebra
This 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.
引文
[1]Sweedler M E. Hopf algebras [M].New York:Benjamin,1969
[2]Abe E. Hopf Algebra [M].London:Cambridge University Press,1980
[3]Montgomery S.Hopf algebras and their actions on rings [J].AMS,1993,82:178-216
[4]Zhang S C.Braided Hopf Algebras [M].Changsha:Hunan Normal University Press,Second Edition
[5]Hungerford T W. Algebra [M].New York:Springer-Verlag,1974
[6]Taft E J,Wilson R L. On antipodes in pointed Hopf algebras [J].J.Alg, 1974,29:27-32
[7]Radford D E. The stucture of Hopf algebras with a projection [J].J. Alg,1985,92:322-347
[8]Shirshov A I. On free Lie rings [J].Mat. sb,1958,2:113-122
[9]Shirshov A I. On bases for free Lie algebra[J].Algebra Logika,1962,1:14-19
[10]Hall M. A basis for free Lie rings and higher commutators in free groups [J]. Proc. Am. Math. Soc,1950,1:575-581
[11]Shirshov A I. Some algorithmic problems for Lie algebras [J].Sib. Mat.Zh, 1962,3:292-296
[12]Belavin A A,Drinfeld V G.On the solutions of the classical Yang-Baxter equa-tions for simple Lie algebras [J].Funct.Anal.Appl.1982,16:1-29
[13]Stolin A. On rationtions of the classical Yang-Baxter equations. Maximal orders in the loop algebra [J].Comm. Math. Phys,1991,141:533-548
[14]Rosso M.An analogue of the Poincare-Birkhoff-Witt theorem and the universal R-matrix of Uq(sl(N+1))[J].Comm. Math. Psys,1989,124:307-318
[15]Yamana H. A Poincare-Birkhoff-Witt theorem for quantized universal envelop-ing algebras of type AN[J].Publ. RIMS.Kyoto Univ,1989,25:503-520
[16]Lusztig G.Canonical bases arising from quantized enveloping algebras [J].J. Am. Math.Soc,1990,3:447-498
[17]Lusztig G.Quivers,perverse sheaves.and quantized enveloping algebras[J].J. Am. Math. Soc,1991,4:365-421
[18]Kashimara M.On crystal bases of the q-analog of universal enveloping algebras [J].Duke Math,1991,63:465-516
[19]Kharchenko V K. An algebra of skew primitive elements [J].Algebra and Logic, 1998,37:101-126
[20]Andruskiewitsch N.Some remarks on Nichols algebras.In Hopf alge-bras:Proccedings from an International Conference Held at DePaul Univer-sity,2003
[21]Andruskiewitsch N, Grana M.Braided Hopf algebras over nonabelian finite groups [J].Bol.Acad.Nac.Cienc.(Cordoba),1999,63:45-78
[22]Grana M.On Nichols algebras of low dimension [J].Amer.Math.Soc,2000,267 :111-134
[23]Kharchenko V.A quantum analog of the Poincare-Birkhoff-Witt theorem [J].Algebra and Logic,1999,38(4):259-276
[24]Lothaire M. Combinatorics on words [M].London:Cambridge University Press,1983
[25]Stern M A.Uber eine zahlentheoretische Funktion [J].J.reine angew. Math,1858,55:193-220
[26]Andruskiewitsch N, Schneider H.J.Finite quantum groups and Cartan matrices [J].Adv.Math,2000,154:1-45
[27]Andruskiewitsch N,Schneider H.-J.Lifting of quantum linear spaces and pointed Hopf algebras of order p3 [J].J.Algebra,1998,209:658-691
[28]Woronowicz S L.Differential calculus on compact matrix pseudogroups(quantum groups)[J].Commun.Math.Phys,1989;122(1):125-170
[29]Nichols W D.Bialgebras of type one[J].Commum.Algebra,1978,6:1521-1552
[30]Ufer S.PBW bases for a class of brained Hopf algebras [J].Journal of Algebra,in press.
[31]Andruskiewitsch N,Schneider H.Pointed Hopf algebras In New Directions in Hopf Algebras [M].London:Cambridge University Press.2000
[32]Andruskiewitsch N. About finite dimension Hopf algebra [J].Amer.Math. Soc,2002,294:1-57
[33]Heckenberger I.Finite dimensional rank 2 Nichols algebras of diagonal type Ⅰ: Examples [J].ArXiv:math.2004
[34]Kharchenko V K.A quantum analog of the Poincare-Birkhoff-Witt theorem [J]. Algebra and Logic,1999,38:259-275
[35]Jacobson N. Lie Algebras [M].New York:Interscience,1962
[36]Cormen T H.Leiserson C E,Rivest R L.Introduction to algorithms [M].MA:MIT Press,1990
[37]Grana M. A freeness theorem for Nichols algebra J].Algebra,2000,231(1):235-257
[38]Lusztig G.Introsuction to quantum groups [M].Boston:Birkhauser,1993
[39]Graham R L,Knuth D E,Patashnik O.Concrete mathematics:a foundation for computer science [M].MA:Addision-Wesley,Reading,1994
[40]Schauenburg P. A characterization of the Borel-like subalgebras of quantum enveloping algebras [J].Commun.Algebra,1996,24:2811-2823
[2]Abe E. Hopf Algebra [M].London:Cambridge University Press,1980
[3]Montgomery S.Hopf algebras and their actions on rings [J].AMS,1993,82:178-216
[4]Zhang S C.Braided Hopf Algebras [M].Changsha:Hunan Normal University Press,Second Edition
[5]Hungerford T W. Algebra [M].New York:Springer-Verlag,1974
[6]Taft E J,Wilson R L. On antipodes in pointed Hopf algebras [J].J.Alg, 1974,29:27-32
[7]Radford D E. The stucture of Hopf algebras with a projection [J].J. Alg,1985,92:322-347
[8]Shirshov A I. On free Lie rings [J].Mat. sb,1958,2:113-122
[9]Shirshov A I. On bases for free Lie algebra[J].Algebra Logika,1962,1:14-19
[10]Hall M. A basis for free Lie rings and higher commutators in free groups [J]. Proc. Am. Math. Soc,1950,1:575-581
[11]Shirshov A I. Some algorithmic problems for Lie algebras [J].Sib. Mat.Zh, 1962,3:292-296
[12]Belavin A A,Drinfeld V G.On the solutions of the classical Yang-Baxter equa-tions for simple Lie algebras [J].Funct.Anal.Appl.1982,16:1-29
[13]Stolin A. On rationtions of the classical Yang-Baxter equations. Maximal orders in the loop algebra [J].Comm. Math. Phys,1991,141:533-548
[14]Rosso M.An analogue of the Poincare-Birkhoff-Witt theorem and the universal R-matrix of Uq(sl(N+1))[J].Comm. Math. Psys,1989,124:307-318
[15]Yamana H. A Poincare-Birkhoff-Witt theorem for quantized universal envelop-ing algebras of type AN[J].Publ. RIMS.Kyoto Univ,1989,25:503-520
[16]Lusztig G.Canonical bases arising from quantized enveloping algebras [J].J. Am. Math.Soc,1990,3:447-498
[17]Lusztig G.Quivers,perverse sheaves.and quantized enveloping algebras[J].J. Am. Math. Soc,1991,4:365-421
[18]Kashimara M.On crystal bases of the q-analog of universal enveloping algebras [J].Duke Math,1991,63:465-516
[19]Kharchenko V K. An algebra of skew primitive elements [J].Algebra and Logic, 1998,37:101-126
[20]Andruskiewitsch N.Some remarks on Nichols algebras.In Hopf alge-bras:Proccedings from an International Conference Held at DePaul Univer-sity,2003
[21]Andruskiewitsch N, Grana M.Braided Hopf algebras over nonabelian finite groups [J].Bol.Acad.Nac.Cienc.(Cordoba),1999,63:45-78
[22]Grana M.On Nichols algebras of low dimension [J].Amer.Math.Soc,2000,267 :111-134
[23]Kharchenko V.A quantum analog of the Poincare-Birkhoff-Witt theorem [J].Algebra and Logic,1999,38(4):259-276
[24]Lothaire M. Combinatorics on words [M].London:Cambridge University Press,1983
[25]Stern M A.Uber eine zahlentheoretische Funktion [J].J.reine angew. Math,1858,55:193-220
[26]Andruskiewitsch N, Schneider H.J.Finite quantum groups and Cartan matrices [J].Adv.Math,2000,154:1-45
[27]Andruskiewitsch N,Schneider H.-J.Lifting of quantum linear spaces and pointed Hopf algebras of order p3 [J].J.Algebra,1998,209:658-691
[28]Woronowicz S L.Differential calculus on compact matrix pseudogroups(quantum groups)[J].Commun.Math.Phys,1989;122(1):125-170
[29]Nichols W D.Bialgebras of type one[J].Commum.Algebra,1978,6:1521-1552
[30]Ufer S.PBW bases for a class of brained Hopf algebras [J].Journal of Algebra,in press.
[31]Andruskiewitsch N,Schneider H.Pointed Hopf algebras In New Directions in Hopf Algebras [M].London:Cambridge University Press.2000
[32]Andruskiewitsch N. About finite dimension Hopf algebra [J].Amer.Math. Soc,2002,294:1-57
[33]Heckenberger I.Finite dimensional rank 2 Nichols algebras of diagonal type Ⅰ: Examples [J].ArXiv:math.2004
[34]Kharchenko V K.A quantum analog of the Poincare-Birkhoff-Witt theorem [J]. Algebra and Logic,1999,38:259-275
[35]Jacobson N. Lie Algebras [M].New York:Interscience,1962
[36]Cormen T H.Leiserson C E,Rivest R L.Introduction to algorithms [M].MA:MIT Press,1990
[37]Grana M. A freeness theorem for Nichols algebra J].Algebra,2000,231(1):235-257
[38]Lusztig G.Introsuction to quantum groups [M].Boston:Birkhauser,1993
[39]Graham R L,Knuth D E,Patashnik O.Concrete mathematics:a foundation for computer science [M].MA:Addision-Wesley,Reading,1994
[40]Schauenburg P. A characterization of the Borel-like subalgebras of quantum enveloping algebras [J].Commun.Algebra,1996,24:2811-2823