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Structure Analysis of the Pohlmeyer-Rehren Lie Algebra and Adaptations of the Hall Algorithm to Non-Free Graded Lie Algebras
The Pohlmeyer-Rehren Lie algebra is an infinite-dimensional -graded Lie algebra that was discovered in the context of string quantization in -dimensional spacetime by K. Pohlmeyer and his collaborators and has more recently been reformulated in terms of the Euler-idempotents of the shuffle Hopf algebra. This thesis is divided into two major parts. In the first part, the structure theory of is discussed. , the stratum of degree zero, is isomorphic to the classical Lie algebra . Now, each stratum is considered as a -module, and a formula for the number of irreducible -modules of each highest weight that occur is given. It is also shown that is not a Kac-Moody algebra.
Based on computer-aided calculations, is conjectured to be generated by the strata of degrees and , but not freely. In an effort to classify the relations, in the second part, the Philip Hall algorithm that iteratively lists (linear) basis elements of a Lie algebra freely generated by a finite set of generators is modified. Any non-free finitely generated Lie algebra can be written as with an ideal encoding the relations. Intended for cases where is not explicitly known, a variant of the algorithm iteratively lists a basis of and a self-reduced basis of . Further modifications that take advantage of restrictions enforced by a gradation on are also given.2021-06-2