Abstract
We review and extend the results about quasicrystal Lie algebras of Patera et al. [Phys. Lett. A 246 (1998) 209], which is a new family of infinite dimensional Lie algebras over the real and complex number fields, whose generators are in a one-to-one correspondence with the points of a one-dimensional quasicrystal. Some new properties of quasicrystal Lie algebras and further details on their representation theory are pointed out and the concept of generalized quasicrystal Lie algebras is presented. The latter allows to associate to the generators of the Lie algebra quasicrystal points of one-dimensional quasicrystals with acceptance windows symmetric around 0, which was not possible in the framework of Patera et al.
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