On the imaginary simple roots of the Borcherds algebra \(\mathfrak g_{\text{II}_{9,1}}\). (English) Zbl 0953.17012
Summary: In a recent paper [O. Bärwald, R. W. Gebert, M. Günaydin and H. Nicolai, Commun. Math. Phys. 195, 29-65 (1998; Zbl 0938.17017)] it was conjectured that the imaginary simple roots of the Borcherds algebra \(g_{\text{II}_{9,1}}\) at level 1 are its only ones. We here propose an independent test of this conjecture, establishing its validity for all roots of norm \(\geqslant-8\). However, the conjecture fails for roots of norm \(-10\) and beyond, as we show by computing the simple multiplicities down to norm \(-24\), which turn out to be remarkably small in comparison with the corresponding \(E_{10}\) multiplicities. Our derivation is based on a modified denominator formula combining the denominator formulas for \(E_{10}\) and \(g_{II_{9,1}}\), and provides an efficient method for determining the imaginary simple roots. In addition, we compute the \(E_{10}\) multiplicities of all roots up to height 231, including levels up to \(\ell=6\) and norms \(-42\).
MSC:
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
Citations:
Zbl 0938.17017References:
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