一類特殊冪零李代數(shù)的結(jié)構(gòu)

胡建華, 趙衛(wèi)萍

(上海理工大學(xué)理學(xué)院,上海 200093)

一類特殊冪零李代數(shù)的結(jié)構(gòu)

胡建華, 趙衛(wèi)萍

(上海理工大學(xué)理學(xué)院,上海 200093)

鑒于冪零李代數(shù)的結(jié)構(gòu)和表示在李理論中有著重要的地位,主要討論復(fù)數(shù)域上一類特殊的6維帶參數(shù)ε的冪零李代數(shù)的代數(shù)結(jié)構(gòu).首先,在同構(gòu)意義下,利用同構(gòu)的定義及性質(zhì),通過大量的推導(dǎo)計算,確定了此類冪零李代數(shù)的自同構(gòu)群同構(gòu)于6階矩陣乘法群;其次,探討了這類冪零李代數(shù)的Centroid代數(shù)的基本性質(zhì),給出了Centroid代數(shù)的矩陣表示,同時得出這類冪零李代數(shù)的Centroid代數(shù)是一個6維冪零李代數(shù);最后,給出了該類冪零李代數(shù)的δ-導(dǎo)子的矩陣表示.特別當(dāng)δ為1時,探討了該類冪零李代數(shù)的導(dǎo)子代數(shù)的結(jié)構(gòu),得出導(dǎo)子代數(shù)是10維李代數(shù),外導(dǎo)子代數(shù)是5維李代數(shù).

李代數(shù);冪零;自同構(gòu);δ-導(dǎo)子

冪零李代數(shù)是李理論中一類重要的李代數(shù),其結(jié)構(gòu)和表示在李理論中有著重要的地位.冪零李代數(shù)因其結(jié)構(gòu)的復(fù)雜性還有很多問題尚未解決,受到許多學(xué)者的關(guān)注,低維冪零李代數(shù)成為學(xué)者們探索的對象.Schneider[1]通過確定基的方法給出了低維冪零李代數(shù)的分類;Graaf[2]利用中心擴張的方法給出了特征不為2的小于等于6維的冪零李代數(shù)的分類;楊恒云等[3]確定了二上同調(diào)群的結(jié)構(gòu).本文在此基礎(chǔ)上,研究復(fù)數(shù)域上一類6維帶參數(shù)的冪零李代數(shù)的代數(shù)結(jié)構(gòu).

1 概 念

定義1[4]設(shè)L是域F上的向量空間,在L中定義了一個李括號積(記為[·,·]),對?x,y∈L,有[x,y]∈L,且以下三個條件成立,稱L為域F上的一個李代數(shù).

a.李括號積是雙線性的;

b.[x,x]=0,?x∈L;

c.[x,[y,z]]+[y,[z,x]]+[z,[x,y]]= 0,?x,y,z∈L(Jacobi等式).

由條件b易得,當(dāng)Char F≠2時,有[x,y]= -[y,x],?x,y∈L.

例1[5]對結(jié)合代數(shù)A,?a,b∈A,定義運算[a,b]=ab-ba,則A構(gòu)成一個李代數(shù).

定義2[6]設(shè)L是域F上的李代數(shù),{Li}(i≥ 0)為L的降中心列.若存在自然數(shù)n∈瓔,使得Ln={0},則稱L為冪零李代數(shù).

采用文獻[2]的記號,L6,21(ε)表示復(fù)數(shù)域瓘上這樣一個6維李代數(shù),其基為{x1,x2,…,x6},其李括號積為

其余的李括號積[xi,xj]=0.這是一個帶參數(shù)ε∈瓘的冪零李代數(shù),且L5={0},x6是中心元,并且由文獻[2]知,對參數(shù)η∈瓘,L6,21(ε)?L6,21(η),當(dāng)且僅當(dāng)存在α∈瓘,使得η=α2ε.本文將刻畫復(fù)數(shù)域上李代數(shù)L6,21(ε)的自同構(gòu)群、Centroid代數(shù)、導(dǎo)子代數(shù)以及δ-導(dǎo)子.

2 自同構(gòu)群

定義3[4,7]設(shè)L是復(fù)數(shù)域瓘上的李代數(shù),若線性變換φ:L→L滿足

則稱φ是L的同態(tài)映射,L的所有同態(tài)映射的集合記為End(L).若φ是可逆線性變換,則稱φ是L的自同構(gòu)映射,L的所有自同構(gòu)映射構(gòu)成一個群,稱為L的自同構(gòu)群,記作Aut L.

定理1 李代數(shù)L6,21(ε)的自同構(gòu)群Aut L6,21(ε)同構(gòu)于6階矩陣乘法群

3 Centroid代數(shù)

定義4[4,10]設(shè)L是復(fù)數(shù)域瓘上的李代數(shù),若線性變換σ∈End(L)且

記E表示6階單位矩陣,Eij表示第i行第j列的元素為1,其余元素均為0的6階矩陣.

定理2 李代數(shù)Cent(L6,21(ε))同構(gòu)于6階矩陣代數(shù)

推論1 李代數(shù)Cent(L6,21(ε))是冪零李代數(shù),且其維數(shù)是6.

證明 由定理2可知,Cent(L6,21(ε))同構(gòu)于矩陣代數(shù)M,而M由對角線元素全相等的下三角矩陣生成,從而M是冪零的,故Cent(L6,21(ε))是冪零的且維數(shù)等于M的維數(shù),為6.

4 δ-導(dǎo)子

定義5[6]設(shè)L是域瓘上的李代數(shù),若線性變換D:L→L,滿足D([x,y])=[D(x),y]+[x,D(y)],?x,y∈L則稱D為L的導(dǎo)子,L的所有導(dǎo)子所成的集合記為Der(L).

性質(zhì)3[6]設(shè)L是域瓘上的李代數(shù),Der(L)是一個李代數(shù).

由文獻[4],對?x∈L,伴隨算子ad x∈Der(L),稱為L的內(nèi)導(dǎo)子.L的所有內(nèi)導(dǎo)子的集合記為Ad(L),構(gòu)成L的內(nèi)導(dǎo)子代數(shù).商代數(shù)Der(L)/ Ad(L)稱為外導(dǎo)子代數(shù).

定義6[13]設(shè)L是域瓘上的李代數(shù),δ∈瓘為任意不為零的數(shù),若線性變換D:L→L,滿足

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(編輯:董 偉)

Structure of a Certain Class of Nilpotent Lie Algebras

HUJianhua, ZHAOWeiping
(College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China)

The structure and representation of nilpotent Lie algebra play an important role in the Lie theory.The algebraic structure of a certain class of six-dimensional nilpotent Lie algebras with the parameterεover the complex field was discussed.It is determined that in the sense of isomorphism,the automorphism group of this class of six-dimensional nilpotent Lie algebra is isomorphic to a six-order matrix multiplication group by using the definition and properties of isomorphism and a large amount of calculation.Then the properties of Centroid algebras of this class of six-dimensional nilpotent Lie algebra were analysed and its matrix representation was given.It is shown that the Centroid algebra is a six-dimensional nilpotent Lie algebra.Finally,the δ-derivation of this class of six-dimensional nilpotent Lie algebras was determined.Especially in the case ofδ=1,the structure of derivation algebras was discussed and it is concluded that the derivation algebra is a ten-dimensional Lie algebra and outer derivation algebra is a fivedimensional Lie algebra.

Lie algebra;nilpotent;automorphism;δ-derivation

O 152

A

1007-6735(2015)03-0215-05

10.13255/j.cnki.jusst.2015.03.003

2014-04-15

國家自然科學(xué)基金資助項目(11201299)

胡建華(1978-),女,講師.研究方向:代數(shù)群及其表示理論.E-mail:smilydragon2011@163.com

??編號:1007-6735(2015)03-0220-05 DOI:10.13255/j.cnki.jusst.2015.03.004