论文标题
Casimir元素和Sugawara的运营商Takiff代数
Casimir elements and Sugawara operators for Takiff algebras
论文作者
论文摘要
对于每一个简单的谎言代数$ \ mathfrak {g} $，我们考虑关联的takiff代数$ \ mathfrak {g}^{} _ {\ ell} $定义为截断的多项式电流lie代数，并在$ \ m athfrak {g} $中使用系数。我们使用$ \ mathfrak {g}^{} _ {\ ell} $的矩阵演示文稿，为通用代数$ {\ rm u}中心的代数独立发电机提供了统一的构造。然后，使用类似的矩阵演示文稿，用于affine kac-moody algebra $ \ wideHat {\ mathfrak {g}}}^{} _ {} _ {\ ell} $，以证明feigin--frenkel theorem theorem theorem theorem描述相应的offerine vertex algebra的中心。证明依赖于lie代数$ \ mathfrak {g}^{} _ {\ ell} $的一组完整的segal-sugawara vectors的明确构造。
For every simple Lie algebra $\mathfrak{g}$ we consider the associated Takiff algebra $\mathfrak{g}^{}_{\ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $\mathfrak{g}$. We use a matrix presentation of $\mathfrak{g}^{}_{\ell}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${\rm U}(\mathfrak{g}^{}_{\ell})$. A similar matrix presentation for the affine Kac--Moody algebra $\widehat{\mathfrak{g}}^{}_{\ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal--Sugawara vectors for the Lie algebra $\mathfrak{g}^{}_{\ell}$.
