## 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

Zbl 0637.00005