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On higher spin realizations of \(K(E_{10})\). (English) Zbl 1342.81191

Summary: Starting from the known unfaithful spinorial representations of the compact subalgebra \(K(E_{10})\) of the split real hyperbolic Kac-Moody algebra \(E_{10}\) we construct new fermionic ‘higher spin’ representations of this algebra (for ‘spin-\(\frac52\)’ and ‘spin-\(\frac72\)’, respectively) in a second quantized framework. Our construction is based on a simplified realization of \(K(E_{10})\) on the Dirac and the vector spinor representations in terms of the associated roots, and on a re-definition of the vector spinor first introduced in [T. Damour and C. Hillmann, J. High Energy Phys. 2009, No. 8, Paper No. 100 (2009)]. The latter replaces manifestly \(\mathrm{SO}(10)\) covariant expressions by new expressions that are covariant w.r.t. \(\mathrm{SO}(1,9)\), the invariance group of the DeWitt metric restricted to the space of scale factors. We present explicit expressions for all \(K(E_{10})\) elements that are associated to real roots of the hyperbolic algebra (of which there are infinitely many), as well as novel explicit realizations of the generators associated to imaginary roots and their multiplicities. We also discuss the resulting realizations of the Weyl group.

MSC:

81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Software:

xAct; GAMMA
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References:

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