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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Higher-order Sugawara operators for affine Lie algebras
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by Roe Goodman and Nolan R. Wallach PDF
Trans. Amer. Math. Soc. 315 (1989), 1-55 Request permission


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 $(\ast )$ \[ [\sigma (u)(\zeta ),X(\eta )] = {c_\infty }D\delta (\zeta /\eta )\sigma ({\nabla _X}u)(\zeta ) + {\text {higher-order}}\;{\text {terms}}\] 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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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 315 (1989), 1-55
  • MSC: Primary 17B67; Secondary 15A72, 20G45
  • DOI:
  • MathSciNet review: 958893