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

Hansen, Nils Bahne

dc.contributor.advisor	Bahns, Dorothea Prof. Dr.
dc.contributor.author	Hansen, Nils Bahne
dc.date.accessioned	2021-06-14T12:51:58Z
dc.date.available	2021-06-20T00:50:13Z
dc.date.issued	2021-06-14
dc.identifier.uri	http://hdl.handle.net/21.11130/00-1735-0000-0008-5858-3
dc.identifier.uri	http://dx.doi.org/10.53846/goediss-8660
dc.language.iso	eng	de
dc.rights.uri	http://creativecommons.org/licenses/by-nc-nd/4.0/
dc.subject.ddc	510	de
dc.title	Structure Analysis of the Pohlmeyer-Rehren Lie Algebra and Adaptations of the Hall Algorithm to Non-Free Graded Lie Algebras	de
dc.type	doctoralThesis	de
dc.contributor.referee	Bahns, Dorothea Prof. Dr.
dc.date.examination	2020-11-18
dc.description.abstracteng	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.	de
dc.contributor.coReferee	Rehren, Karl-Henning Prof. Dr.
dc.subject.eng	Pohlmeyer-Rehren Lie algebra	de
dc.subject.eng	Lie algebra of truncated tensors	de
dc.subject.eng	infinite-dimensional Lie algebra	de
dc.subject.eng	graded Lie algebra	de
dc.subject.eng	shuffle Hopf algebra	de
dc.subject.eng	Eulerian idempotent	de
dc.subject.eng	Lyndon words	de
dc.subject.eng	Kac-Moody algebra	de
dc.subject.eng	representation theory	de
dc.subject.eng	so(n,C)-modules	de
dc.subject.eng	representations of the complex special orthogonal Lie algebras	de
dc.subject.eng	weight space decomposition	de
dc.subject.eng	Clebsch-Gordan problem	de
dc.subject.eng	string quantization	de
dc.subject.eng	Pohlmeyer approach	de
dc.subject.eng	Nambu-Goto string	de
dc.subject.eng	Meusburger-Rehren quantization	de
dc.subject.eng	Poisson algebra of invariant charges of the Nambu-Goto string	de
dc.subject.eng	Pohlmeyer Poisson algebra	de
dc.subject.eng	exceptional elements	de
dc.subject.eng	Phillip Hall algorithm	de
dc.subject.eng	non-free Lie algebra	de
dc.subject.eng	pseudo-Hall-basis	de
dc.subject.eng	pseudo-Hall-exhaustibility	de
dc.subject.eng	Mathematica code	de
dc.identifier.urn	urn:nbn:de:gbv:7-21.11130/00-1735-0000-0008-5858-3-1
dc.affiliation.institute	Fakultät für Mathematik und Informatik	de
dc.subject.gokfull	Mathematics (PPN61756535X)	de
dc.description.embargoed	2021-06-20
dc.identifier.ppn	1760430579

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