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Higher-order Sugawara operators for affine Lie algebras. (English) Zbl 0676.17013

Let \(\hat {\mathfrak g}\) be the affine Lie algebra associated to a finite- dimensional complex semisimple Lie algebra \({\mathfrak g}\), i.e. \(\hat{\mathfrak g}\) is the universal one-dimensional central extension of the Lie algebra of maps \(S^ 1\to {\mathfrak g}\) given by Laurent polynomials. The well- known Sugawara construction [H. Sugawara, A field theory of currents, Phys. Rev. 170, 1659-1662 (1968)] constructs operators \(\{L_ n\}_{n\geq 0}\) as formal infinite sums of products of pairs of elements of \(\hat {\mathfrak g}\). The \(L_ n\) make sense as operators on certain representations of \(\hat {\mathfrak g}\) (e.g. highest weight representations) and form a representation of the Virasoro algebra: \[ [L_ n,L_ m]=((m^ 3-m)/12)\delta_{n,-m}c \] for some \(c\in {\mathbb{C}}\) (depending on the representation); they also intertwine correctly with the operators of \(\hat{\mathfrak g}:\) \[ (*)\quad [L_ n,X\zeta^ m]= mX\zeta^{m+n}\quad (x\in {\mathfrak g},\quad \zeta \in S^ 1). \] This paper is an attempt to generalize these results to higher order products of elements of \(\hat{\mathfrak g}\). A linear map \(\sigma\) from the \({\mathfrak g}\)-invariant elements of the symmetric algebra \(S({\mathfrak g})\) to formal operator fields \(X(\zeta)\) is constructed, such that the following analogue of (*) holds: \[ [\sigma (u)(\zeta),X(\eta)]=cD\delta (\zeta /\eta)\sigma (\nabla_ Xu)(\zeta)+\text{ higher order terms.} \] If u is the quadratic Casimir element in \(S({\mathfrak g})\), then \(\sigma(u)\) is the formal operator \(\sum L_ n\zeta^ n\) and we recover the previous situation, since the “higher order terms” then vanish. This vanishing is also proved when \({\mathfrak g}={\mathfrak sl}_ n({\mathbb{C}})\) and u runs over a particular choice of generators of the invariants, and for arbitrary \({\mathfrak g}\) when \(\deg u\leq 4.\)
There is now a dichotomy according to whether \(c\in {\mathbb{C}}\) is zero or not. If \(c\neq 0\), one recovers the fact that there is a compatible action of the Virasoro algebra on the representation. If \(c=0\), and if the vanishing property occurs, one has a family of operators which commute with the action of \(\hat{\mathfrak g}\). They provide a substitute for the centre of \(U(\hat{\mathfrak g})\), which is uninteresting, being generated by the central element of \(\hat{\mathfrak g}\) [V. Chari and S. Ilangovan, J. Algebra 90, 476-490 (1984; Zbl 0545.17003)]. This is used to obtain the commutant, composition series and character formulas for certain irreducible representations on which \(c=0\). This generalizes earlier work of the second author in the \({\mathfrak sl}_ 2\) case [N. Wallach, Math. Z. 196, 303-313 (1987; Zbl 0637.17011)].
Reviewer: A.N.Pressley

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
81T99 Quantum field theory; related classical field theories
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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