量子环面上无限维李代数的结构
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摘要
扩张仿射李代数(简写为EALA)是一类重要的李代数,它包含了我们熟知的有限型和仿射型Kac-Moody代数.这类李代数最初由文[H-KT]的作者引进研究,并被称为拟单李代数.随后,文[BGK]的作者对单边情形拟单李代数进行了详细研究.在文[AABGP]中,作者引进半格的概念,并对扩张仿射根系进行了分类,同时也给出了扩张仿射李代数的具体实现.和仿射Kac-Moody代数不同的是,扩张仿射李代数的坐标代数不但可以是多变量罗朗多项式环,还可以是某些特殊的交错代数,若当代数以及量子环面代数.
量子环面代数作为罗朗多项式代数的非交换化的推广,已被很多学者进行了研究,如[P],[KPS],[McP]等.从量子环面C_q和它的导子代数Der(C_q)出发,可以构造很多重要的李代数.例如,在文[BGK]中,作者运用量子环面和Der(C_q)的一个由部分特殊导子组成的子代数实现了A型的扩张仿射李代数.本文从两个变量的量子环面C_q出发,讨论了C_q的斜导子李代数以及斜导子李代数的一类扩张得到的无限维李代数的结构性质.
设R~2是二维欧氏空间,Z~2是R~2中的一个格.由映射f:Z~2×Z~2→C~*,f(a,b)=q~(a2b1-a1b2),a=(a1,a2),b=(b1,b2)∈Z~2,确定的Z~2的子加群rad(f)={a∈Z~2|f(a,b)=1,(?)b∈Z~2}称为f的根基.设q∈C~*,C_q:=C_q[x_1~(±1),x_2~(±1)]是结合于矩阵(?)的二秩量子环面,它是含单位元1的结合代数.C_q的导子代数Der(C_q)的一个子代数B=span_C{x~a(a2d1-a1d2),(0,0)≠a∈rad(f)}(?) span_C{adx~r,r∈Z~2,r(?)rad(f)}称为C_q的斜导子李代数,其中d_2=(?),i=1,2是C_q的度导子.当q=1和q不是单位根时,B分别同构于Virasoro-like代数和Virasoro-like代数的q-类似代数.作李代数的半直积(?)(q)=B(?)C_q,其中C_q是(?)(q)的理想.这个李代数可以看成是一阶Heisenberg-Virasoro代数的二阶推广.记L(q)=[(?)(q),(?)(q)],我们有(?)(q)=L(q)(?)C,其中C是(?)(q)的中心.本文主要讨论了当q≠1是p次本原单位根时,李代数L(q)的自同构群,导子代数,泛覆盖以及李代数B的Leibniz中心扩张和不变对称双线性型.具体内容组织如下:
在第一章中,我们首先确定了李代数L(q)的自同构群,并由此得到L(q)的自同构群.主要结论是:该结果推广了[XLT],[ZhT]的结果.
在第二章,我们计算了L(q)到它的伴随模的导子,结论为:Der(L(q))=Innder(L(q))(?) Outder(L(q)),其中dim_C Outder(L(q))=5.
在第三章,我们给出了L(q)的泛覆盖(?),它是由L(q)的四个线性无关的C值2-上循环确定的四维覆盖中心扩张.
在第四章中,我们计算了斜导子李代数召关于平凡模C的Leibniz二上同调群HL~2(B,C),发现该群与B的二上同调群相同,具体结论为HL~2(B,C)=CB_1+CB_2.接着证明了B的所有不变对称双线性型组成的向量空间S(B,C)是一维的,运用这个结果和文[HPL]的一个推论,我们又直接推出HL~2(B,C)=H~2(B,C).
The extended affine Lie algebras (EALA for short) are an important class of Lie algebras which includes the well-known Kac-Moody Lie algebras of finite and affine type. These kinds of Lie algebras were first introduced in the paper [H-KT], and called the quasisimple Lie algebras. In [BGK], the authors studied the quasisimple Lie algebras of simply laced cases. In the book [AABGP], the authors used the concept of a semilattice to classify the extended affine root systems, and they providedconcrete realization of EALA's as well. Different from the affine Kac-Moody Lie algebras,the coordinate algebras of EALA's not only can be the Laurent polynomial algebras of multiple variables, but also can be alternative algebras, Jordan algebras and the quantum torus.
The quantum torus, which are the non-commutative generalization of the Laurent polynomial rings, have been studied by many researchers, such as [P],[KPS],[McP]. From the quantum torus C_q and its derivation algebra Der(C_q), one can construct many important Lie algebras. For instance, the authors in [BGK] realized the EALA's of type A by using the quantum torus along with Lie subalgebras of Der(C_q). In this thesis, we mainly discuss the structure properties of some infinite dimensional Lie algebras, which are the extension of the skew derivation Lie algebras arising from the quantum torus in two variables.
Let R~2 be the 2-dimensional Euclidean space. Z~2 is a lattice in R~2.Let f: Z~2×Z~2→C~* be the map defined by f(a,b) = q~(a_2b_1-a_1b_2),where a = (a_1,a_2),b= (b_1,b_2)∈Z~2.The additive subgroup rad(f)={a∈Z~2|f(a,b)= l,(?)b∈Z~2} is called the radical of f.Let q∈C~*,and C_q:=C_q[(?),(?)] be the quantum torusof rank two associated to the matrix (?).It is an associative algebra withunity. The subalgebra of Der(C_q)B=span_C{x~a(a_2d_1-a_1d_2),(0,0)≠a∈rad(f)}(?)span_C{adx~r,r∈Z~2,r(?)rad(f)}is called the skew derivation Lie algebra over C_q,where d_i,i=1,2 are the degree derivations. If q=1 or q is not a root of unity, then B is isomorphic to the Virasoro-like algebra or the q-analog of the Virasoro-like algebra respectively. Let (?)(q) = B(?)C_q,the semi-direct product of the Lie algebras B and C_q,where C_q is an ideal of (?)(q).This Lie algebra can be viewed as a generalization of the rank-one Heisenberg-Virasoro algebra. Denote L(q)=[(?)(q),(?)(q)], then we have (?)(q)=L(q)(?)C,where C is the center of (?)(q).In this thesis, I first determine the automorphism group, the derivation algebra, the covering of the Lie algebra L(q), where q≠1 is a p-th primitive root of unity. Next, I compute the Leibniz second cohomology group as well as the invariant symmetric bilinear form of B. The main contents are arranged as follows:
In chapter one, we determine the automorphism group of L(q), we get thatAs a corollary, we obtain the automorphism group of (?)(q). These results generalize those obtained in [XLT] and [ZhT].
In chapter two, we compute the derivations from L{q) to its adjoint module L(q).The result is Der(L(q))=Innder(L(q)) (?) Outder(L(q)), where Outder(L(q)) is of 5-dimensional.
In chapter three, we give the universal covering (?) of L(q) by showing that the covering central extension (?) is determined by four linearly independent 2-cocycles of L(q) with values in C.
In chapter four, we study the Leibniz second cohomology group H L~2(B, C) of the skew derivaion Lie algebra B, and find that it is equal to the second cohomology group of B. Moreover, we show that the vector space consisting of the invariant symmetric bilinear forms of B is one dimensional. As a corollary, we obtain that H L~2(B, C) = H~2(B,C) by applying a result in [HPL].
量子环面代数作为罗朗多项式代数的非交换化的推广,已被很多学者进行了研究,如[P],[KPS],[McP]等.从量子环面C_q和它的导子代数Der(C_q)出发,可以构造很多重要的李代数.例如,在文[BGK]中,作者运用量子环面和Der(C_q)的一个由部分特殊导子组成的子代数实现了A型的扩张仿射李代数.本文从两个变量的量子环面C_q出发,讨论了C_q的斜导子李代数以及斜导子李代数的一类扩张得到的无限维李代数的结构性质.
设R~2是二维欧氏空间,Z~2是R~2中的一个格.由映射f:Z~2×Z~2→C~*,f(a,b)=q~(a2b1-a1b2),a=(a1,a2),b=(b1,b2)∈Z~2,确定的Z~2的子加群rad(f)={a∈Z~2|f(a,b)=1,(?)b∈Z~2}称为f的根基.设q∈C~*,C_q:=C_q[x_1~(±1),x_2~(±1)]是结合于矩阵(?)的二秩量子环面,它是含单位元1的结合代数.C_q的导子代数Der(C_q)的一个子代数B=span_C{x~a(a2d1-a1d2),(0,0)≠a∈rad(f)}(?) span_C{adx~r,r∈Z~2,r(?)rad(f)}称为C_q的斜导子李代数,其中d_2=(?),i=1,2是C_q的度导子.当q=1和q不是单位根时,B分别同构于Virasoro-like代数和Virasoro-like代数的q-类似代数.作李代数的半直积(?)(q)=B(?)C_q,其中C_q是(?)(q)的理想.这个李代数可以看成是一阶Heisenberg-Virasoro代数的二阶推广.记L(q)=[(?)(q),(?)(q)],我们有(?)(q)=L(q)(?)C,其中C是(?)(q)的中心.本文主要讨论了当q≠1是p次本原单位根时,李代数L(q)的自同构群,导子代数,泛覆盖以及李代数B的Leibniz中心扩张和不变对称双线性型.具体内容组织如下:
在第一章中,我们首先确定了李代数L(q)的自同构群,并由此得到L(q)的自同构群.主要结论是:该结果推广了[XLT],[ZhT]的结果.
在第二章,我们计算了L(q)到它的伴随模的导子,结论为:Der(L(q))=Innder(L(q))(?) Outder(L(q)),其中dim_C Outder(L(q))=5.
在第三章,我们给出了L(q)的泛覆盖(?),它是由L(q)的四个线性无关的C值2-上循环确定的四维覆盖中心扩张.
在第四章中,我们计算了斜导子李代数召关于平凡模C的Leibniz二上同调群HL~2(B,C),发现该群与B的二上同调群相同,具体结论为HL~2(B,C)=CB_1+CB_2.接着证明了B的所有不变对称双线性型组成的向量空间S(B,C)是一维的,运用这个结果和文[HPL]的一个推论,我们又直接推出HL~2(B,C)=H~2(B,C).
The extended affine Lie algebras (EALA for short) are an important class of Lie algebras which includes the well-known Kac-Moody Lie algebras of finite and affine type. These kinds of Lie algebras were first introduced in the paper [H-KT], and called the quasisimple Lie algebras. In [BGK], the authors studied the quasisimple Lie algebras of simply laced cases. In the book [AABGP], the authors used the concept of a semilattice to classify the extended affine root systems, and they providedconcrete realization of EALA's as well. Different from the affine Kac-Moody Lie algebras,the coordinate algebras of EALA's not only can be the Laurent polynomial algebras of multiple variables, but also can be alternative algebras, Jordan algebras and the quantum torus.
The quantum torus, which are the non-commutative generalization of the Laurent polynomial rings, have been studied by many researchers, such as [P],[KPS],[McP]. From the quantum torus C_q and its derivation algebra Der(C_q), one can construct many important Lie algebras. For instance, the authors in [BGK] realized the EALA's of type A by using the quantum torus along with Lie subalgebras of Der(C_q). In this thesis, we mainly discuss the structure properties of some infinite dimensional Lie algebras, which are the extension of the skew derivation Lie algebras arising from the quantum torus in two variables.
Let R~2 be the 2-dimensional Euclidean space. Z~2 is a lattice in R~2.Let f: Z~2×Z~2→C~* be the map defined by f(a,b) = q~(a_2b_1-a_1b_2),where a = (a_1,a_2),b= (b_1,b_2)∈Z~2.The additive subgroup rad(f)={a∈Z~2|f(a,b)= l,(?)b∈Z~2} is called the radical of f.Let q∈C~*,and C_q:=C_q[(?),(?)] be the quantum torusof rank two associated to the matrix (?).It is an associative algebra withunity. The subalgebra of Der(C_q)B=span_C{x~a(a_2d_1-a_1d_2),(0,0)≠a∈rad(f)}(?)span_C{adx~r,r∈Z~2,r(?)rad(f)}is called the skew derivation Lie algebra over C_q,where d_i,i=1,2 are the degree derivations. If q=1 or q is not a root of unity, then B is isomorphic to the Virasoro-like algebra or the q-analog of the Virasoro-like algebra respectively. Let (?)(q) = B(?)C_q,the semi-direct product of the Lie algebras B and C_q,where C_q is an ideal of (?)(q).This Lie algebra can be viewed as a generalization of the rank-one Heisenberg-Virasoro algebra. Denote L(q)=[(?)(q),(?)(q)], then we have (?)(q)=L(q)(?)C,where C is the center of (?)(q).In this thesis, I first determine the automorphism group, the derivation algebra, the covering of the Lie algebra L(q), where q≠1 is a p-th primitive root of unity. Next, I compute the Leibniz second cohomology group as well as the invariant symmetric bilinear form of B. The main contents are arranged as follows:
In chapter one, we determine the automorphism group of L(q), we get thatAs a corollary, we obtain the automorphism group of (?)(q). These results generalize those obtained in [XLT] and [ZhT].
In chapter two, we compute the derivations from L{q) to its adjoint module L(q).The result is Der(L(q))=Innder(L(q)) (?) Outder(L(q)), where Outder(L(q)) is of 5-dimensional.
In chapter three, we give the universal covering (?) of L(q) by showing that the covering central extension (?) is determined by four linearly independent 2-cocycles of L(q) with values in C.
In chapter four, we study the Leibniz second cohomology group H L~2(B, C) of the skew derivaion Lie algebra B, and find that it is equal to the second cohomology group of B. Moreover, we show that the vector space consisting of the invariant symmetric bilinear forms of B is one dimensional. As a corollary, we obtain that H L~2(B, C) = H~2(B,C) by applying a result in [HPL].
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