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Representations of nine-dimensional Levi decomposition Lie algebras and Lie-Einstein Spaces in 7 Dimension

Khanal, Sunil
http://orcid.org/0000-0002-9388-4938
http://rave.ohiolink.edu/etdc/view?acc_num=toledo1595699071800903

Abstract Details

2020, Doctor of Philosophy, University of Toledo, Mathematics.
This dissertation is all about Real matrix representations of nine-dimensional Levi decomposition Lie algebras and Lie-Einstein Spaces in 7 Dimension. To obtain a matrix representation for each of the indecomposable 9-dimensionalreal Lie algebras that have a non-trivial Levi decomposition, I have used nine different techniques. There are 65 of such algebras. Given a real Lie algebra L of dimension n a well- known theorem due to Ado asserts that L has a faithful representation as a subalgebra of gl(n,R) for somen. For second part of my dissertation, I have been working on invariant metrics on Lie groups. More precisely I have been constructing automorphisms of a Lie group and using them to reduce the metric and then looking at various curvature properties. I have been looking at Lie groups of dimensions six and seven. Even though the metrics are reduced the formulas for Ricci and sectional curvature are still very complicated and difficult to analyze in general. I have been able to locate a few special interesting cases where the metric can be dramatically simplified. The ultimate goal is to find Einstein spaces, that is, metrics for which the Ricci tensor is proportional to the metric. After reducing the metric using some automorphisms of the Lie algebra, I try to solve the conditions interactively with Maple. The conditions typically involve both the entries in the metric and parameters in the Lie algebra. Of course, in dimensions six and seven the conditions are extremely complicated, so I have to make strong assumptions on the form of the metric, for example, that the metric is diagonal in the Lie algebra basis.
Gerard Thompson (Committee Chair)
Akaki Tikaradze (Committee Member)
Ekaterina Shemyakova (Committee Member)
Ryad Ghanam (Committee Member)
129 p.
Mathematics
Lie Algebra; Lie group

Recommended Citations

Citations

  • Khanal, S. (2020). Representations of nine-dimensional Levi decomposition Lie algebras and Lie-Einstein Spaces in 7 Dimension [Doctoral dissertation, University of Toledo]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1595699071800903

    APA Style (7th edition)

  • Khanal, Sunil. Representations of nine-dimensional Levi decomposition Lie algebras and Lie-Einstein Spaces in 7 Dimension. 2020. University of Toledo, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=toledo1595699071800903.

    MLA Style (8th edition)

  • Khanal, Sunil. "Representations of nine-dimensional Levi decomposition Lie algebras and Lie-Einstein Spaces in 7 Dimension." Doctoral dissertation, University of Toledo, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1595699071800903

    Chicago Manual of Style (17th edition)

toledo1595699071800903
197
© 2020, some rights reserved.Creative Commons License Representations of nine-dimensional Levi decomposition Lie algebras and Lie-Einstein Spaces in 7 Dimension by Sunil Khanal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by University of Toledo and OhioLINK.