Research Outputs

Permanent URI for this communityhttps://hdl.handle.net/20.500.14288/2

Browse

Filters

2010 - 201922020 - 20231

Advanced Search

Filter by

Settings

Department: search.filters.department.Department of PhysicsSubject: search.filters.subject.Applied mathematics

Search Results

Now showing 1 - 3 of 3
  • Placeholder
    PublicationMetadata only
    ‘Anti-commutable’ local pre-Leibniz algebroids and admissible connections
    (Elsevier, 2023) Department of Physics; N/A; Dereli, Tekin; Doğan, Keremcan; Faculty Member; PhD Student; Department of Physics; College of Sciences; Graduate School of Sciences and Engineering; 201358; N/A
    The concept of algebroid is convenient as a basis for constructions of geometrical frameworks. For example, metric-affine and generalized geometries can be written on Lie and Courant algebroids, respectively. Furthermore, string theories might make use of many other algebroids such as metric algebroids, higher Courant algebroids, or conformal Courant algebroids. Working on the possibly most general algebroid structure, which generalizes many of the algebroids used in the literature, is fruitful as it creates a chance to study all of them at once. Local pre-Leibniz algebroids are such general ones in which metric-connection geometries are possible to construct. On the other hand, the existence of the 'locality operator', which is present for the left-Leibniz rule for the bracket, necessitates the modification of torsion and curvature operators in order to achieve tensorial quantities. In this paper, this modification of torsion and curvature is explained from the point of view that the modification is applied to the bracket instead. This leads one to consider 'anti-commutable' local pre-Leibniz algebroids which satisfy an anti-commutativity-like property defined with respect to a choice of an equivalence class of connections. These 'admissible' connections are claimed to be the necessary ones while working on a geometry of algebroids. This claim is due to the fact that one can prove many desirable properties and relations if one uses only admissible connections. For instance, for admissible connections, we prove the first and second Bianchi identities, Cartan structure equations, Cartan magic formula, the construction of Levi-Civita connections, the decomposition of connection in terms of torsion and non-metricity. These all are possible because the modified bracket becomes anti-symmetric for an admissible connection so that one can apply the machinery of almost-or pre-Lie algebroids. We investigate various algebroid structures from the literature and show that they admit admissible connections which are metric-compatible in some generalized sense. Moreover, we prove that local pre-Leibniz algebroids that are not anti-commutable cannot be equipped with a torsion-free, and in particular Levi-Civita, connection.
  • Placeholder
    PublicationMetadata only
    Degenerate spin groups as semi-direct products
    (Springer, 2010) Kocak, Sahin; Limoncu, Murat; Department of Physics; Dereli, Tekin; Faculty Member; Department of Physics; College of Sciences; 201358
    Let Q be a symmetric bilinear form on R(n)=R(p+q+r) with corank r, rank p+q and signature type (p, q), p resp. q denoting positive resp. negative dimensions. We consider the degenerate spin group Spin(Q) = Spin(p, q, r) in the sense of Crumeyrolle and prove that this group is isomorphic to the semi-direct product of the nondegenerate and indefinite spin group Spin(p, q) with the additive matrix group Mat (p + q, r)
  • Placeholder
    PublicationMetadata only
    Maximal linear subspaces of strong self-dual 2-forms and the Bonan 4-form
    (Elsevier, 2011) Bilge, Ayşe Humeyra; Koçak, Şahin; Department of Physics; Dereli, Tekin; Faculty Member; Department of Physics; College of Sciences; 201358
    The notion of self-duality of 2-forms in 4-dimensions plays an eminent role in many areas of mathematics and physics, but although the 2-forms have a genuine meaning related to curvature and gauge-field-strength in higher dimensions also, their "self-duality" is something which is almost avoided above 4-dimensions. We show that self-duality of 2-forms is a very natural notion in higher (even) dimensions also and we prove the equivalence of some scattered and rarely used definitions in the literature. We demonstrate the usefulness of this higher self-duality by studying it in 8-dimensions and we derive a natural expression for the Bonan form in terms of self-dual 2-forms and we give an explicit expression of the local action of SO(8) on the Bonan form. (C) 2010 Elsevier Inc. All rights reserved.