Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Higher-order Sugawara operators for affine Lie algebras
HTML articles powered by AMS MathViewer

by Roe Goodman and Nolan R. Wallach PDF
Trans. Amer. Math. Soc. 315 (1989), 1-55 Request permission

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 $(\ast )$ \[ [\sigma (u)(\zeta ),X(\eta )] = {c_\infty }D\delta (\zeta /\eta )\sigma ({\nabla _X}u)(\zeta ) + {\text {higher-order}}\;{\text {terms}}\] 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 $\hat {\mathfrak {g}}$.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B67, 15A72, 20G45
  • Retrieve articles in all journals with MSC: 17B67, 15A72, 20G45
Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 315 (1989), 1-55
  • MSC: Primary 17B67; Secondary 15A72, 20G45
  • DOI: https://doi.org/10.1090/S0002-9947-1989-0958893-5
  • MathSciNet review: 958893