Structure Analysis of the Pohlmeyer-Rehren Lie Algebra and Adaptations of the Hall Algorithm to Non-Free Graded Lie Algebras

by Nils Bahne Hansen
Doctoral thesis
Date of Examination:2020-11-18
Date of issue:2021-06-14
Advisor:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Karl-Henning Rehren
Persistent Address: http://dx.doi.org/10.53846/goediss-8660

 

 

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Abstract

English

The Pohlmeyer-Rehren Lie algebra $\mathfrak{g}$ is an infinite-dimensional $\mathbb{Z}$-graded Lie algebra that was discovered in the context of string quantization in $d$-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 $\mathfrak{g}$ is discussed. $\mathfrak{g}_0$, the stratum of degree zero, is isomorphic to the classical Lie algebra $\mathfrak{so}(d,\mathbb{C})$. Now, each stratum is considered as a $\mathfrak{g}_0$-module, and a formula for the number of irreducible $\mathfrak{g}_0$-modules of each highest weight that occur is given. It is also shown that $\mathfrak{g}$ is not a Kac-Moody algebra. Based on computer-aided calculations, $\mathfrak{g}$ is conjectured to be generated by the strata of degrees $0$ and $1$, 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 $L(X)$ freely generated by a finite set of generators $X$ is modified. Any non-free finitely generated Lie algebra can be written as $L(X)/I$ with an ideal $I$ encoding the relations. Intended for cases where $I$ is not explicitly known, a variant of the algorithm iteratively lists a basis of $L(X)/I$ and a self-reduced basis of $I$. Further modifications that take advantage of restrictions enforced by a gradation on $L(X)/I$ are also given.
Keywords: Pohlmeyer-Rehren Lie algebra; Lie algebra of truncated tensors; infinite-dimensional Lie algebra; graded Lie algebra; shuffle Hopf algebra; Eulerian idempotent; Lyndon words; Kac-Moody algebra; representation theory; so(n,C)-modules; representations of the complex special orthogonal Lie algebras; weight space decomposition; Clebsch-Gordan problem; string quantization; Pohlmeyer approach; Nambu-Goto string; Meusburger-Rehren quantization; Poisson algebra of invariant charges of the Nambu-Goto string; Pohlmeyer Poisson algebra; exceptional elements; Phillip Hall algorithm; non-free Lie algebra; pseudo-Hall-basis; pseudo-Hall-exhaustibility; Mathematica code
 
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