**维拉宿代數**（Virasoro algebra）是單位圓上微分算子所組成的李代數的中心拓展（英语：），在複數域上的無限維李代數。這與仿射Kac-Moody代數（英语：）關係密切（參看Sugawara構造）。Virasoro 代數的么正表示描繪兩維共形場論的對稱性。

定義
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**维拉宿代數**是一李代數，生成元是

* {\displaystyle {L\_{n}:n\in \mathbb {Z} }},
* c ，
* 符合：{\displaystyle [L\_{m},L\_{n}]=(m-n)L\_{m+n}+\delta \_{m+n}{\frac {(m^{3}-m)}{12}}c}

推导
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维拉宿代數可以被认为是以下Witt 代数（英语：） 的 中心拓展（英语：）:

{\displaystyle [l\_{m},l\_{n}]=(m-n)l\_{m+n}},

{\displaystyle [{\bar {l}}\_{m},{\bar {l}}\_{n}]=(m-n){\bar {l}}\_{m+n}},

{\displaystyle [l\_{m},{\bar {l}}\_{n}]=0}.

对于一李代数{\displaystyle \ {\bf {g}}}, 其在复数域{\displaystyle \ {\bf {C}}}的 central extension{\displaystyle \ {\tilde {g}}} 满足下列交换子:

{\displaystyle [{\tilde {x}},{\tilde {y}}]\_{\tilde {g}}=[x,y]\_{g}+cp(x,y),}

{\displaystyle [{\tilde {x}},c]\_{\tilde {g}}=0,}

{\displaystyle [c,c]\_{\tilde {g}}=0,}

其中{\displaystyle \ {\tilde {x}},{\tilde {y}}\in {\tilde {g}},x,y\in g,c\in {\bf {C}},p:{\tilde {g}}\times {\tilde {g}}\rightarrow {\bf {C}}}. 由此定义, 维拉宿代數的生成元满足以下交换子

{\displaystyle [L\_{m},L\_{n}]=(m-n)L\_{m+n}+cp(m,n)}.

{\displaystyle p(m,n)}可以由以下条件决定:

* 交换子必须是反对易的, 所以{\displaystyle p(m,n)=-p(n,m)}
* 可以观察到, 如果定义以下生成元

{\displaystyle {\hat {L}}\_{n}=L\_{n}+{\frac {cp(n,0)}{n}},n\neq 0}

{\displaystyle {\hat {L}}\_{0}=L\_{0}+{\frac {cp(1,-1)}{2}},}

它们满足

{\displaystyle [{\hat {L}}\_{n},{\hat {L}}\_{0}]=nL\_{n}+cp(n,0)=n{\hat {L}}\_{n},}

{\displaystyle [{\hat {L}}\_{1},{\hat {L}}\_{-1}]=2L\_{0}+cp(1,-1)=2{\hat {L}}\_{0}.}

比较函数{\displaystyle p(m,n)}的定义可以得知,{\displaystyle p(1,-1)}与{\displaystyle p(n,0)}总是可以被设为0.

* 交换子满足雅可比恒等式,即

{\displaystyle \ 0=[[L\_{m},L\_{n}],L\_{0}]+[[L\_{n},L\_{0}],L\_{m}]+[[L\_{0},L\_{m}],L\_{n}]}

|  |  |
| --- | --- |
|  | {\displaystyle \ =(m-n)cp(m+n,0)+ncp(n,m)-mcp(m,n)} |
|  | {\displaystyle \ =(m+n)p(n,m)} |

所以{\displaystyle p(n,m)=0}如果{\displaystyle n\neq -m}, 即唯一的非零 central extension为{\displaystyle p(n,-n)}且{\displaystyle |n|>=2}.

* 最后计算以下雅克比恒等式

{\displaystyle \ 0=[[L\_{-n+1},L\_{n}],L\_{-1}]+[[L\_{n},L\_{-1}],L\_{-n+1}]+[[L\_{-1},L\_{-n+1}],L\_{n}]}

|  |
| --- |
| {\displaystyle \ =(-2n+1)cp(1,-1)+(n+1)cp(n-1,-n+1)+(n-1)cp(-n,n)} |
|  |

可知{\displaystyle p(m,n)}满足以下递推公式

|  |
| --- |
| {\displaystyle \ p(n,-n)={\frac {n+1}{n-2}}p(n-1,-n+1)} |
| {\displaystyle \ ={\frac {n+1}{n-2}}{\frac {n}{n-3}}p(n-2,-n+2)}=... |
| {\displaystyle \ ={\frac {n+1}{n-2}}{\frac {n}{n-3}}...{\frac {4}{1}}p(2,-2)} |
| {\displaystyle \ ={n+1 \choose 3}{\frac {1}{2}}} |
| {\displaystyle \ ={\frac {1}{12}}(n+1)n(n-1),} |
|  |

其中归一化条件为{\displaystyle p(2,-2)={\frac {1}{2}}}.综上所述, Witt algebra在复数域唯一非零的central extension, 即维拉宿代数的生成元满足以下交换子

{\displaystyle [L\_{m},L\_{n}]=(m-n)L\_{m+n}+c{\frac {1}{12}}(n+1)n(n-1)\delta \_{m+n,0}}.

局部保角变换
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群表示论
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Verma模
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超对称维拉宿代数
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註
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參考
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* V.G. Kac: "*Infinite dimensional Lie algebras*", Cambridge University Press
* V.G. Kac / A.K. Raina : "*Bombay Lectures on highest weight representations*" , World Scientific, Singapore
* Di Francesco / Mathieu / Senechal : "Conformal field theory", Springer Verlag
* Wakimoto: "*Infinite-dimensional Lie algebras*" （日語書《無限次元環》的譯本）, American Mathematical Society
* Ralph Blumenhagen/ Erik Plauschinn : "Introduction to conformal field theory: with applications to string theory", Springer Lecture notes in physics 779, Page 15