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112 constructions of the basic representation of the loop group of \(E_ 8\). (English) Zbl 0642.17013

Anomalies, geometry, topology, Symp. Chicago/Ill. 1985, 276-298 (1985).
[For the entire collection see Zbl 0637.00005.]
In this report, we describe a natural family of vertex constructions of the basic representation of the affine Kac-Moody algebra \(\hat{\mathfrak g}\) associated to a simple finite-dimensional Lie algebra \({\mathfrak g}\) of type \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\) or \(E_ 8\). Namely, we show that maximal Heisenberg subalgebras of \(\hat{\mathfrak g}\) are parametrized, up to conjugacy, by conjugacy classes of elements of the Weyl group W of \({\mathfrak g}\). Given \(w\in W\), let \(\hat s_ w\) denote the associated Heisenberg subalgebra of \(\hat{\mathfrak g}\), and let \(\tilde S_ w\) denote the centralizer of \(\hat s_ w\) in the loop group \(\tilde G\) of the simply-connected group G whose Lie algebra is \({\mathfrak g}\). We show that the basic representation \((V,\pi_ 0)\) of \(\hat{\mathfrak g}\) remains irreducible under the pair \((\hat s_ w,\tilde S_ w)\). This leads to a vertex construction of V, so that for \(w=1\) (resp. \(w=Coxeter\) element) we recover the homogeneous (resp. principal) realization; for \(w=-1\) we recover the known construction. Thus, to each conjugacy class of W we associate canonically a vertex realization of the basic representation of \(\hat{\mathfrak g}\). In particular, in the case of \(\hat E_ 8\) we obtain 112 such constructions.
The homogeneous realization of \(\hat E_ 8\) plays an important role in the construction of the heterotic string. We hope that the large variety of constructions of \(\hat E_ 8\) provided by this paper could be useful for the treatment of various symmetry breaking patterns.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties

Citations:

Zbl 0637.00005