Representations of affine truncations of representation involutive-semirings of Lie algebras and root systems of higher type

• Main Author: Graves, Timothy W
• Other Authors: Gannon, Terry (Mathematical and Statistical Sciences)
• Format: Others
• Language: en
• Published: 2011
• Online Access: http://hdl.handle.net/10048/1897

An important component of a rational conformal field theory is a representation of a certain involutive-semiring. In the case of Wess-Zumino-Witten models, the involutive-semiring is an affine truncation of the representation involutive-semiring of a finite-dimensional semisimple Lie algebra. We show how root systems naturally correspond to representations of an affine truncation of the representation involutive-semiring of sl_2(C). By reversing this procedure, one can, to a representation of an affine truncation of the representation involutive-semiring of an arbitrary finite-dimensional semisimple Lie algebra, associate a root system and a Cartan matrix of higher type. We show how this same procedure applies to a special subclass of representations of the nontruncated representation involutive-semiring, leading to higher-type analogues of the affine Cartan matrices. Finally, we extend the known results for sl_3(C) with a symmetric classification, a construction, and a list of computer-generated examples. === Mathematics