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As explained in [1], the di culties of understanding hyperbolic Kac{Moody algebras on the one hand and Borcherds algebras (also called generalized Kac{Moody algebras [3]) on the other hand are to some extent complementary: while a Lorentzian Kac{Moody algebra has a well-understood root system, but the structure of the algebra and its root spaces (including their dimensions!) is very complicated, Borcherds algebras may possess a simple realization in terms of physical string states, but usually haveavery complicated root system due to the appearance of imaginary (i.e. non-positive norm) simple roots. The Chevalley generators corresponding to imaginary simple roots of gII9;1 are needed to complete the subalgebra E10 to the full Lie algebra of physical states. This can be seen by decomposing the vector space M: = gII9;1 E10 (2) into an in nite direct sum of \missing modules " all of which are highest or lowest weight modules w.r.t. the subalgebra E10 (see [1, 11]). This implies that all of gII9;1 can be generated from the highest and lowest weight states by the action of (i.e. multiple commutation with) the E10 raising or lowering operators. However, not all the lowest weight states in M correspond to imaginary simple roots of gII9;1. This is because the commutation of two lowest weight states yields another lowest weight state; yet it is only those lowest weight states which cannot be obtained as multiple commutators of previous states and which must therefore be added \by hand"

Year: 2011

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