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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Higher-order Sugawara operators for affine Lie algebras

Authors: Roe Goodman and Nolan R. Wallach
Journal: Trans. Amer. Math. Soc. 315 (1989), 1-55
MSC: Primary 17B67; Secondary 15A72, 20G45
MathSciNet review: 958893
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Abstract: Let $ \hat{\mathfrak{g}}$ be the affine Lie algebra associated to a simple Lie algebra $ \mathfrak{g}$. Representations of $ \hat{\mathfrak{g}}$ are described by current fields $ X(\zeta)$ on the circle $ {\mathbf{T}}\;(X \in \mathfrak{g}$ and $ \zeta \in {\mathbf{T}})$. In this paper a linear map $ \sigma $ from the symmetric algebra $ S(\mathfrak{g})$ to (formal) operator fields on a suitable category of $ \hat{\mathfrak{g}}$ modules is constructed. The operator fields corresponding to $ \mathfrak{g}$-invariant elements of $ S(\mathfrak{g})$ are called Sugawara fields. It is proved that they satisfy commutation relations of the form $ (\ast)$

$\displaystyle [\sigma (u)(\zeta),X(\eta)] = {c_\infty }D\delta (\zeta /\eta)\sigma ({\nabla _X}u)(\zeta) + {\text{higher-order}}\;{\text{terms}}$

with the current fields, where $ {c_\infty }$ is a renormalization of the central element in $ \hat{\mathfrak{g}}$ and $ D\delta $ is the derivative of the Dirac delta function. The higher-order terms in $ (\ast)$ are studied using results from invariant theory and finite-dimensional representation theory of $ \mathfrak{g}$. For suitably normalized invariants $ u$ of degree $ 4$ or less, these terms are shown to be zero. This vanishing is also proved for $ \mathfrak{g} = {\text{sl}}(n,{\mathbf{C}})$ and $ u$ running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever $ {c_\infty }$ is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for $ \hat{\mathfrak{g}}$.

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