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Bergander, Philip.
Twisted derivations, *quasi*-hom-*Lie* algebras and their *quasi*-deformations.

Degree: Culture and Communication, 2017, Mälardalen University

URL: http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-35553

Quasi-hom-Lie algebras (qhl-algebras) were introduced by Larsson and Silvestrov (2004) as a generalisation of hom-Lie algebras, which are a deformation of Lie algebras. Lie algebras are defined by an operation called bracket, [·,·], and a three-term Jacobi identity. By the theorem from Hartwig, Larsson, and Silvestrov (2003), this bracket and the three-term Jacobi identity are deformed into a new bracket operation, <·,·>, and a six-term Jacobi identity, making it a quasi-hom-Lie algebra. Throughout this thesis we deform the Lie algebra sl_{2}(F), where F is a field of characteristic 0. We examine the quasi-deformed relations and six-term Jacobi identities of the following polynomial algebras: F[t], F[t]/(t^{2}), F[t]/(t^{3}), F[t]/(t^{4}), F[t]/(t^{5}), F[t]/(t^{n}), where n is a positive integer ≥2, and F[t]/((t-t_{0})^{3}). Larsson and Silvestrov (2005) and Larsson, Sigurdsson, and Silvestrov (2008) have already examined some of these cases, which we repeat for the reader's convenience. We further investigate the following σ-twisted derivations, and how they act in the different cases of mentioned polynomial algebras: the ordinary differential operator, the shifted difference operator, the Jackson q-derivation operator, the continuous q-difference operator, the Eulerian operator, the divided difference operator, and the nilpotent imaginary derivative operator. We also introduce a new, general, σ-twisted derivation operator, which is σ(t) as a polynomial of degree k.

Subjects/Keywords: quasi-hom-Lie algebra; hom-Lie algebra; Lie algebra; Twisted derivation; sigma derivation; quasi-Lie algebra; derivation; quasi deformation; quasi-deformation; twisted vector field; polynomial algebra; quotient ring; Algebra and Logic; Algebra och logik

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APA (6^{th} Edition):

Bergander, P. (2017). Twisted derivations, quasi-hom-Lie algebras and their quasi-deformations. (Thesis). Mälardalen University. Retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-35553

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Bergander, Philip. “Twisted derivations, quasi-hom-Lie algebras and their quasi-deformations.” 2017. Thesis, Mälardalen University. Accessed January 29, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-35553.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Bergander, Philip. “Twisted derivations, quasi-hom-Lie algebras and their quasi-deformations.” 2017. Web. 29 Jan 2020.

Vancouver:

Bergander P. Twisted derivations, quasi-hom-Lie algebras and their quasi-deformations. [Internet] [Thesis]. Mälardalen University; 2017. [cited 2020 Jan 29]. Available from: http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-35553.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Bergander P. Twisted derivations, quasi-hom-Lie algebras and their quasi-deformations. [Thesis]. Mälardalen University; 2017. Available from: http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-35553

Not specified: Masters Thesis or Doctoral Dissertation