SwePub - sökning: WFRF:(Arnlind Joakim)

Numrering	Referens	Omslagsbild	Hitta
1.	Al-Shujary, Ahmed, 1980- (författare) Kähler-Poisson Algebras 2018 Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract The focus of this thesis is to introduce the concept of Kähler-Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds. We first give here a review of the geometry of Kähler manifolds and Lie-Rinehart algebras. After that we give the definition and basic properties of Kähler-Poisson algebras. It is then shown that the Kähler type condition has consequences that allow for an identification of geometric objects in the algebra which share several properties with their classical counterparts. Furthermore, we introduce a concept of morphism between Kähler-Poisson algebras and show its consequences. Detailed examples are provided in order to illustrate the novel concepts.
2.	Al-Shujary, Ahmed, 1980- (författare) Kähler-Poisson Algebras 2020 Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract In this thesis, we introduce Kähler-Poisson algebras and study their basic properties. The motivation comes from differential geometry, where one can show that the Riemannian geometry of an almost Kähler manifold can be formulated in terms of the Poisson algebra of smooth functions on the manifold. It turns out that one can identify an algebraic condition in the Poisson algebra (together with a metric) implying that most geometric objects can be given a purely algebraic formulation. This leads to the definition of a Kähler-Poisson algebra, which consists of a Poisson algebra and a metric fulfilling an algebraic condition. We show that every Kähler- Poisson algebra admits a unique Levi-Civita connection on its module of inner derivations and, furthermore, that the corresponding curvature operator has all the classical symmetries. Moreover, we present a construction procedure which allows one to associate a Kähler-Poisson algebra to a large class of Poisson algebras. From a more algebraic perspective, we introduce basic notions, such as morphisms and subalgebras, as well as direct sums and tensor products. Finally, we initiate a study of the moduli space of Kähler-Poisson algebras; i.e for a given Poisson algebra, one considers classes of metrics giving rise to non-isomorphic Kähler-Poisson algebras. As it turns out, even the simple case of a Poisson algebra generated by two variables gives rise to a nontrivial classification problem.
3.	Arnlind, Joakim, et al. (författare) A noncommutative catenoid 2018 Ingår i: Letters in Mathematical Physics. - : Springer Netherlands. - 0377-9017 .- 1573-0530. ; 108:7, s. 1601-1622 Tidskriftsartikel (refereegranskat)abstract A noncommutative algebra corresponding to the classical catenoid is introduced together with a differential calculus of derivations. We prove that there exists a unique metric and torsion-free connection that is compatible with the complex structure, and the curvature is explicitly calculated. A noncommutative analogue of the fact that the catenoid is a minimal surface is studied by constructing a Laplace operator from the connection and showing that the embedding coordinates are harmonic. Furthermore, an integral is defined and the total curvature is computed. Finally, classes of left and right modules are introduced together with constant curvature connections, and bimodule compatibility conditions are discussed in detail.
4.	Arnlind, Joakim, et al. (författare) Affine transformation crossed product type algebras and noncommutative surfaces 2009. - 503 Ingår i: Operator structures and dynamical systems. - Providence, Rhode Island : American Mathematical Society (AMS). - 9780821847473 - 9780821881828 ; , s. 1-25 Bokkapitel (refereegranskat)abstract Several classes of -algebras associated to teh action of an affine transformation are considered, and an investigation of the interplay between the different classes is initiated. Connections are established that relate representations of -algebras, geometry of algebraic surfaces, dynamics of affine transformations, graphs and algebras coming from a quantization procedure of Poisson structures. In particular, algebras related to surgaced being inverse images of fourth order polynomials (in ) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.
5.	Arnlind, Joakim, et al. (författare) An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces 2016 Ingår i: Nonlinear Analysis. - : PERGAMON-ELSEVIER SCIENCE LTD. - 0362-546X .- 1873-5215. ; 134, s. 70-104 Tidskriftsartikel (refereegranskat)abstract We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on ( ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach. (C) 2015 Elsevier Ltd. All rights reserved.
6.	Arnlind, Joakim, et al. (författare) Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras 2011 Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 52:12, s. 123502- Tidskriftsartikel (refereegranskat)abstract As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3653197]
7.	Arnlind, Joakim (författare) Curvature and geometric modules of noncommutative spheres and tori 2014 Ingår i: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 55:4, s. 041705- Tidskriftsartikel (refereegranskat)abstract When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connection from Euclidean space, and we compute its scalar curvature.
8.	Arnlind, Joakim, et al. (författare) Deformed noncommutative tori 2012 Ingår i: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 53:7, s. 073505- Tidskriftsartikel (refereegranskat)abstract We recall a construction of non-commutative algebras related to a one-parameter family of (deformed) spheres and tori, and show that in the case of tori, the -algebras can be completed into C-algebras isomorphic to the standard non-commutative torus. As the former was constructed in the context of matrix (or fuzzy) geometries, it provides an important link to the framework of non-commutative geometry, and opens up for a concrete way to study deformations of non-commutative tori. Furthermore, we show how the well-known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of finite-dimensional representations of the deformed algebras, and their projective modules are described together with connections of constant curvature.
9.	Arnlind, Joakim, et al. (författare) Discrete minimal surface algebras 2010 Ingår i: Symmetry, Integrability and Geometry. - 1815-0659. ; 6, s. Paper 042,18- Tidskriftsartikel (refereegranskat)abstract We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d <= 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is ( generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
10.	Arnlind, Joakim (författare) Eigenvalue dynamics and membrane solutions 2004 Licentiatavhandling (övrigt vetenskapligt/konstnärligt)
11.	Arnlind, Joakim, et al. (författare) Eigenvalue dynamics, Follytons and large N limits of matrices 2006 Ingår i: Applications of Random Matrices in Physics. - DORDRECHT : SPRINGER. - 1402045298 - 9781402045295 - 9781402045318 ; , s. 89-94 Konferensbidrag (refereegranskat)abstract How do the eigenvalues of a “free” hermitian N × N matrix X(t) evolve in time? The answer is provided by the rational Calogero-Moser systems [5, 13] if (!) the initial conditions are chosen such that i[X(0),Ẋ(0)] has a non-zero eigenvalue of multiplicity N–1; for generic X(0),Ẋ(0) the question remained unanswered for 30 years.
12.	Arnlind, Joakim, et al. (författare) Eigenvalue Dynamics off the Calogero-Moser system 2004 Ingår i: Letters in Mathematical Physics. - : Springer. - 0377-9017 .- 1573-0530. ; 68:2, s. 121-129 Tidskriftsartikel (refereegranskat)abstract By finding N(N-1)/2 suitable conserved quantities, free motions of real symmetric N x N matrices X(t), with arbitrary initial conditions, are reduced to nonlinear equations involving only the eigenvalues of X-in contrast to the rational Calogero-Moser system, for which [X(0), X(0)] has to be purely imaginary, of rank one.
13.	Arnlind, Joakim, et al. (författare) Fuzzy Riemann surfaces 2009 Ingår i: Journal of High Energy Physics (JHEP). - : Springer Science and Business Media LLC. - 1126-6708 .- 1029-8479. ; :6, s. 047- Tidskriftsartikel (refereegranskat)abstract We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C (onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.
14.	Arnlind, Joakim, et al. (författare) Goldfish geodesics and hamiltonian reduction of matrix dynamics 2008 Ingår i: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 84:1, s. 89-98 Tidskriftsartikel (refereegranskat)abstract We describe the Hamiltonian reduction of a time-dependent real-symmetric NxN matrix system to free vector dynamics, and also provide a geodesic interpretation of Ruijsenaars-Schneider systems. The simplest of the latter, the goldfish equation, is found to represent a flat-space geodesic in curvilinear coordinates.
15.	Arnlind, Joakim, 1979- (författare) Graph Techniques for Matrix Equations and Eigenvalue Dynamics 2008 Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract One way to construct noncommutative analogues of a Riemannian manifold Σ is to make use of the Toeplitz quantization procedure. In Paper III and IV, we construct C-algebras for a continuously deformable class of spheres and tori, and by introducing the directed graph of a representation, we can completely characterize the representation theory of these algebras in terms of the corresponding graphs. It turns out that the irreducible representations are indexed by the periodic orbits and N-strings of an iterated map s:(reals) 2→(reals)2 associated to the algebra. As our construction allows for transitions between spheres and tori (passing through a singular surface), one easily sees how the structure of the matrices changes as the topology changes. In Paper II, noncommutative analogues of minimal surface and membrane equations are constructed and new solutions are presented -- some of which correspond to minimal tori embedded in S7. Paper I is concerned with the problem of finding differential equations for the eigenvalues of a symmetric N × N matrix satisfying Xdd=0. Namely, by finding N(N-1)/2 suitable conserved quantities, the time-evolution of X (with arbitrary initial conditions), is reduced to non-linear equations involving only the eigenvalues of Χ.
16.	Arnlind, Joakim, et al. (författare) Kahler-Poisson algebras 2019 Ingår i: Journal of Geometry and Physics. - : ELSEVIER SCIENCE BV. - 0393-0440 .- 1879-1662. ; 136, s. 156-172 Tidskriftsartikel (refereegranskat)abstract We introduce Kahler-Poisson algebras as analogues of algebras of smooth functions on Kahler manifolds, and prove that they share several properties with their classical counterparts on an algebraic level. For instance, the module of inner derivations of a Kahler-Poisson algebra is a finitely generated projective module, and allows for a unique metric and torsion-free connection whose curvature enjoys all the classical symmetries. Moreover, starting from a large class of Poisson algebras, we show that every algebra has an associated Kahler-Poisson algebra constructed as a localization. At the end, detailed examples are provided in order to illustrate the novel concepts. (C) 2018 Elsevier B.V. All rights reserved.
17.	Arnlind, Joakim (författare) Levi-Civita connections for a class of noncommutative minimal surfaces 2021 Ingår i: International Journal of Geometric Methods in Modern Physics (IJGMMP). - : WORLD SCIENTIFIC PUBL CO PTE LTD. - 0219-8878. ; 18:12 Tidskriftsartikel (refereegranskat)abstract In this paper, we study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e. finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for which there is a natural concept of torsion, and we prove that there exist metric and torsion free connections for every minimal surface in this class.
18.	Arnlind, Joakim, et al. (författare) Levi-Civita Connections on Quantum Spheres 2022 Ingår i: Mathematical physics, analysis and geometry. - : Springer. - 1385-0172 .- 1572-9656. ; 25:3 Tidskriftsartikel (refereegranskat)abstract We introduce q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics. We also give metric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.
19.	Arnlind, Joakim (författare) Low Dimensional Matrix Representations for Noncommutative Surfaces of Arbitrary Genus 2020 Ingår i: Mathematical physics, analysis and geometry. - : SPRINGER. - 1385-0172 .- 1572-9656. ; 23:2 Tidskriftsartikel (refereegranskat)abstract In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated in detail and graph representations are used in order to understand the structure of non-zero matrix elements. In particular, for arbitrary genus greater than one, we explicitly construct classes of irreducible two and three dimensional representations. The existence of representations crucially depends on the analytic structure of the polynomial defining the surface as a level set in Double-struck capital R-3.
20.	Arnlind, Joakim, et al. (författare) Multi-linear Formulation of Differential Geometry and Matris Regularizations 2012 Ingår i: Journal of differential geometry. - : International Press. - 0022-040X .- 1945-743X. ; 91:1, s. 1-39 Tidskriftsartikel (refereegranskat)abstract We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi-linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingartens formula, the Ricci curvature, and the Codazzi-Mainardi equations. For matrix analogues of embedded surfaces, we define discrete curvatures and Euler characteristics, and a non-commutative Gauss-Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and explicit examples are provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.
21.	Arnlind, Joakim, et al. (författare) Noncommutative minimal embeddings and morphisms of pseudo-Riemannian calculi 2021 Ingår i: Journal of Geometry and Physics. - : ELSEVIER. - 0393-0440 .- 1879-1662. ; 159 Tidskriftsartikel (refereegranskat)abstract In analogy with classical submanifold theory, we introduce morphisms of real metric calculi together with noncommutative embeddings. We show that basic concepts, such as the second fundamental form and the Weingarten map, translate into the noncommutative setting and, in particular, we prove a noncommutative analogue of Gauss equations for the curvature of a submanifold. Moreover, the mean curvature of an embedding is readily introduced, giving a natural definition of a noncommutative minimal embedding, and we illustrate the novel concepts by considering the noncommutative torus as a minimal surface in the noncommutative 3-sphere. (c) 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
22.	Arnlind, Joakim, et al. (författare) Noncommutative Minimal Surfaces 2016 Ingår i: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 106:8, s. 1109-1129 Tidskriftsartikel (refereegranskat)abstract We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass representation.
23.	Arnlind, Joakim, et al. (författare) Noncommutative Riemann Surfaces by Embeddings in R-3 2009 Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 288:2, s. 403-429 Tidskriftsartikel (refereegranskat)abstract We introduce C-Algebras of compact Riemann surfaces ∑ as non-commutative analogues of the Poisson algebra of smooth functions on ∑. Representations of these algebrasgive rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.
24.	Arnlind, Joakim (författare) Noncommutative Riemannian geometry of Kronecker algebras 2024 Ingår i: Journal of Geometry and Physics. - : ELSEVIER. - 0393-0440 .- 1879-1662. ; 199 Tidskriftsartikel (refereegranskat)abstract We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on metrics and bimodule connections compatible with the *-structure of the algebra. As an illustration, these concepts are applied to the noncommutative torus where examples of torsion free and metric (Levi-Civita) connections are given. In the main part of the paper, noncommutative geometric aspects of (generalized) Kronecker algebras are considered. The structure of derivations and differential calculi is explored, and torsion free bimodule connections are studied together with their compatibility with hermitian forms, playing the role of metrics on the module of differential forms. Moreover, for several different choices of Lie algebras of derivations, non -trivial Levi-Civita connections are constructed. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
25.	Arnlind, Joakim, et al. (författare) On the Chern-Gauss-Bonnet theorem for the noncommutative 4-sphere 2017 Ingår i: JOURNAL OF GEOMETRY AND PHYSICS. - : ELSEVIER SCIENCE BV. - 0393-0440. ; 111, s. 126-141 Tidskriftsartikel (refereegranskat)abstract We construct a differential calculus over the noncommutative 4-sphere in the framework of pseudo-Riemannian calculi, and show that for every metric in a conformal class of perturbations of the round metric, there exists a unique metric and torsion-free connection. Furthermore, we find a localization of the projective module corresponding to the space of vector fields, which allows us to formulate a Chern-Gauss-Bonnet type theorem for the noncommutative 4-sphere. (C) 2016 Elsevier B.V. All rights reserved.
26.	Arnlind, Joakim, et al. (författare) On the construction of fuzzy spaces and modules over shift algebras 2023 Ingår i: Advances in Theoretical and Mathematical Physics. - : INT PRESS BOSTON, INC. - 1095-0761 .- 1095-0753. ; 27:4, s. 1223-1274 Tidskriftsartikel (refereegranskat)abstract We introduce shift algebras as certain crossed product algebras based on general function spaces and study properties, as well as the classification, of a particular class of modules depending on a set of matrix parameters. It turns out that the structure of these modules depends in a crucial way on the properties of the function spaces. Moreover, for a class of subalgebras related to compact manifolds, we provide a construction procedure for the corresponding fuzzy spaces, i.e. sequences of finite dimensional modules of increasing dimension as the deformation parameter tends to zero, as well as infinite dimensional modules related to fuzzy non -compact spaces.
27.	Arnlind, Joakim, et al. (författare) Projections, modules and connections for the noncommutative cylinder 2020 Ingår i: Advances in Theoretical and Mathematical Physics. - : INT PRESS BOSTON, INC. - 1095-0761 .- 1095-0753. ; 24:3, s. 527-562 Tidskriftsartikel (refereegranskat)abstract We initiate a study of projections and modules over a noncommutative cylinder, a simple example of a noncompact noncommutative manifold. Since its algebraic structure turns out to have many similarities with the noncommutative torus, one can develop several concepts in a close analogy with the latter. In particular, we exhibit a countable number of nontrivial projections in the algebra of the noncommutative cylinder itself, and show that they provide concrete representatives for each class in the corresponding K-0 group. We also construct a class of bimodules endowed with connections of constant curvature. Furthermore, with the noncommutative cylinder considered from the perspective of pseudo-Riemannian calculi, we derive an explicit expression for the Levi-Civita connection and compute the Gaussian curvature.
28.	Arnlind, Joakim, et al. (författare) Pseudo-Riemannian Geometry in Terms of Multi-Linear Brackets 2014 Ingår i: Letters in Mathematical Physics. - : Springer Verlag (Germany). - 0377-9017 .- 1573-0530. ; 104:12, s. 1507-1521 Tidskriftsartikel (refereegranskat)abstract We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kahler geometry.
29.	Arnlind, Joakim (författare) Representation theory of C-algebras for a higher order class of spheres and tori 2008 Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 49:5, s. 053502-1-053502-13 Tidskriftsartikel (refereegranskat)abstract We construct C-algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of "loop" and "string" representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the "Henon algebras," for which the dynamical map is a generalized Henon map, and give an example where irreducible representations of all dimensions exist.
30.	Arnlind, Joakim, et al. (författare) Riemannian curvature of the noncommutative 3-sphere 2017 Ingår i: Journal of Noncommutative Geometry. - : EUROPEAN MATHEMATICAL SOC. - 1661-6952 .- 1661-6960. ; 11:2, s. 507-536 Tidskriftsartikel (refereegranskat)abstract In order to investigate to what extent the calculus of classical (pseudo-) Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civitas theorem, which states that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivating example, we consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and explicitly determine the torsion-free and metric connection, as well as the curvature operator together with its scalar curvature.
31.	Arnlind, Joakim, et al. (författare) Spinning Membranes 2004 Ingår i: Physics Letters B. - : Elsevier BV. - 0370-2693 .- 1873-2445. ; 599:1-2, s. 118-128 Tidskriftsartikel (refereegranskat)abstract We present new solutions of the classical equations of motion of bosonic (matrix-)membranes. Those relating to minimal surfaces in spheres provide spinning membrane solutions in AdS(p) X S-q, as well as in flat space-time. Nontrivial reductions of the BMN matrix model equations are also given.
32.	Arnlind, Joakim, et al. (författare) Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras 2014 Ingår i: Springer Proceedings in Mathematics and Statistics. - Berlin, Heidelberg : Springer. - 9783642553608 - 9783642553615 ; , s. 123-144 Konferensbidrag (refereegranskat)abstract The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced 3-Lie algebras
33.	Arnlind, Joakim, et al. (författare) Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras 2010 Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 51:4, s. 043515-11 Tidskriftsartikel (refereegranskat)abstract The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom-Nambu-Lie algebras obtained using this construction.
34.	Arnlind, Joakim, et al. (författare) The world as quantized minimal surfaces 2013 Ingår i: Physics Letters B. - : Elsevier. - 0370-2693 .- 1873-2445. ; 723:4-5, s. 397-400 Tidskriftsartikel (refereegranskat)abstract It is pointed out that the equations less thanbrgreater than less thanbrgreater thanSigma(d)(i=1)[X-i, [X-i, X-j]] = 0 less thanbrgreater than less thanbrgreater than(and its super-symmetrizations, playing a central role in M-theory matrix models) describe non-commutative minimal surfaces - and can be solved as such.
35.	Arnlind, Joakim, et al. (författare) Trace Extensions, Determinant Bundles, and Gauge Group Cocycles 2002 Ingår i: Letters in Mathematical Physics. - : Springer. - 0377-9017 .- 1573-0530. ; 62:2, s. 101-110 Tidskriftsartikel (refereegranskat)abstract We study the geometry of determinant line bundles associated with Dirac operators on compact odd-dimensional manifolds. Physically, these arise as (local) vacuum line bundles in quantum gauge theory. We give a simplified derivation of the commutator anomaly formula using a construction based on noncyclic trace extensions and associated nonmultiplicative renormalized determinants.
36.	Bäck, Per, 1987- (författare) On Hom-associative Ore Extensions 2022 Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract In this thesis, we introduce and study hom-associative Ore extensions. These are non-unital, non-associative, non-commutative polynomial rings in which the associativity condition is “twisted” by an additive group homomorphism. In particular, they are examples of hom-associative algebras, and they generalize the classical non-commutative polynomial rings introduced by Ore known as Ore extensions to the non-unital, hom-associative setting. At the same time, when the twisted associativity condition is null, they also generalize to the general non-unital, non-associative setting. We deduce necessary and sufficient conditions for hom-associative Ore extensions to exist, and construct concrete examples thereof. These include hom-associative generalizations of the quantum plane, the universal enveloping algebra of the two-dimensional non-abelian Lie algebra, and the first Weyl algebra, to name a few. The aforementioned algebras turn out to be formal hom-associative deformations of their associative counterparts, the latter two which cannot be formally deformed in the associative setting. Moreover, these are all weakly unital algebras, and we provide a way of embedding any multiplicative, non-unital hom-associative algebra into a multiplicative, weakly unital hom-associative algebra. This generalizes the classical unitalization of non-unital, associative algebras. We then study the hom-associative Weyl algebras in arbitrary characteristic, classify them up to isomorphism, and in the zero characteristic case, we prove that an analogue of the Dixmier conjecture is true. We also study hom-modules over hom-associative rings, and by doing so, we are able to prove a Hilbert's basis theorem for hom-associative Ore extensions. Our theorem includes as special cases both the classical Hilbert's basis theorem for Ore extensions and a Hilbert's basis theorem for unital, non-associative Ore extensions. Last, we construct examples of both hom-associative and non-associative Ore extensions which are all Noetherian by our theorem.
37.	Ilwale, Kwalombota, 1980- (författare) Noncommutative Riemannian Geometry of Twisted Derivations 2023 Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract A twisted derivation is a generalized derivative satisfying a twisted version of the ordinary Leibniz rule for products. In particular, a (σ, τ )-derivation on an algebra A, is a derivation where Leibniz rule is twisted by two endomorphisms σ and τ on A. Such derivations play an important role in the theory of quantum groups, as well as in the context of discretized and deformed derivatives. In this thesis, we develop a (commutative and noncommutative) differential geometry based on (σ, τ )- derivations. To this end, we introduce the notion of (σ, τ )-algebra, consisting of an associative algebra together with a set of (σ, τ )-derivations, to construct connections satisfying a twisted Leibniz rule in analogy with (σ, τ )-derivations. We show that such connections always exist on projective modules and that it is possible to construct connections compatible with a hermitian form. To construct torsion and curvature of (σ, τ )-connections, we introduce the notion of (σ, τ )-Lie algebra and demonstrate that it is possible to construct a Levi-Civita (σ, τ )-connection. Having constructed the framework for studying (σ, τ)-connections, we demonstrate that the framework applied to commutative algebras can help to also give a good understanding of (σ, τ )-derivations on commutative algebras. In particular, we introduce a notion of symmetric (σ, τ )-derivations together with some regularity conditions. For example, we show that strongly regular (σ, τ )-derivations are always inner and there exist a symmetric (σ, τ )-connection on symmetric (σ, τ )- algebras. Finally, we introduce a (σ, τ)-Hochschild cohomology theory which in first degree captures the outer (σ, τ )-derivations of an associative algebra. Along the way, examples including both commutative and noncommutative algebras are presented to illustrate the novel concepts.
38.	Kardell, Marcus (författare) New Phenomena in the World of Peaked Solitons 2016 Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract The aim of this work is to present new contributions to the theory of peaked solitons. The thesis consists of two papers,which are named “Newsolutionswith peakon creation in the Camassa–HolmandNovikov equations” and “Peakon-antipeakon solutions of the Novikov equation” respectively.In Paper I, a new kind of peakon-like solution to the Novikov equation is discovered, by transforming the one-peakon solution via a Lie symmetry transformation. This new kind of solution is unbounded as x → +∞ and/or x → –∞, and has a peak, though only for some interval of time. Thus, the solutions exhibit creation and/or destruction of peaks. We make sure that the peakon-like function is still a solution in the weak sense for those times where the function is non-differentiable. We find that similar solutions, with peaks living only for some interval in time, are validweak solutions to the Camassa–Holm equation, though it appears that these can not be obtained via a symmetry transformation.In Paper II we investigate multipeakon solutions of the Novikov equation, in particular interactions between peakons with positive amplitude and antipeakons with negative amplitude. The solutions are given by explicit formulas, which makes it possible to analyze them in great detail. As in the Camassa–Holm case, the slope of the wave develops a singularity when a peakon collides with an antipeakon, while the wave itself remains continuous and can be continued past the collision to provide a global weak solution. However, the Novikov equation differs in several interesting ways from other peakon equations, especially regarding asymptotics for large times. For example, peakons and antipeakons both travel to the right, making it possible for several peakons and antipeakons to travel together with the same speed and collide infinitely many times. Such clusters may exhibit very intricate periodic or quasi-periodic interactions. It is also possible for peakons to have the same asymptotic velocity but separate at a logarithmic rate; this phenomenon is associated with coinciding eigenvalues in the spectral problem coming from the Lax pair, and requires nontrivial modifications to the previously known solution formulas which assume that all eigenvalues are simple.
39.	Ongong'a, Elvice, 1990- (författare) Classification and Construction of Low-dimensional Hom-Lie Algebras and Ternary Hom-Nambu-Lie Algebras 2021 Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract This thesis concerns the construction and classification of low-dimensional Hom-Lie algebras and ternary Hom-Nambu-Lie algebras. A classification of 3-dimensional Hom-Lie algebras is given for nilpotent linear endomorphism, as a twisting map, and a construction of 4-dimensional Hom-Lie algebras is done. Results on the dimension of the space of endomorphisms that turn a skew-symmetric algebra into a Hom-Lie algebra are also given in this thesis. A class of 3-dimensional ternary Hom-Nambu-Lie algebras with nilpotent linear maps are constructed and classified.In Chapter 2, we derive conditions for an arbitrary n-dimensional algebra to be a Hom-Lie algebra, in the form of a system of polynomial equations, containing both structure constants of the skew-symmetric bilinear map and constants describing the twisting linear endomorphism. When the algebra is 3 or 4-dimensional, we describe the realisation of Hom-Lie algebras when the dimension of the space of such linear endomorphisms, as vector spaces, is minimum. For the 3-dimensional case we give all possible families of 3-dimensional Hom-Lie algebras arising from a general nilpotent linear endomorphism constructed up to isomorphism together with non-isomorphic canonical representatives for all the families in that case. We further give a list of 4-dimensional Hom-Lie algebras arising from general nilpotent linear endomorphisms.In Chapter 3, we describe the dimension of the space of possible linear endomorphisms that turn skew-symmetric three-dimensional algebras into Hom-Lie algebras. We find a correspondence between the rank of a matrix containing the structure constants of the bilinear product and the dimension of the space of Hom-Lie structures. Examples from classical complex Lie algebras are given to demonstrate this correspondence.In Chapter 4, the space of possible Hom-Lie structures on complex 4-dimensional Lie algebras is considered in terms of linear maps that turn the Lie algebras into Hom-Lie algebras. Hom-Lie structures and automorphism groups on the representatives of isomorphism classes of complex 4-dimensional Lie algebras are described.In Chapter 5, we construct ternary Hom-Nambu-Lie algebras from Hom-Lie algebras through a process known as induction. The induced algebras are constructed from a class of Hom-Lie algebra with nilpotent linear map. The families of ternary Hom-Nambu-Lie arising in this way of construction are classified for a given class of nilpotent linear maps. In addition, some results giving conditions on when morphisms of Hom-Lie algebras can still remain morphisms for the induced ternary Hom-Nambu-Lie algebras are given.
40.	Tiger Norkvist, Axel, 1994- (författare) Morphisms of real calculi from a geometric and algebraic perspective 2021 Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract Noncommutative geometry has over the past four of decades grown into a rich field of study. Novel ideas and concepts are rapidly being developed, and a notable application of the theory outside of pure mathematics is quantum theory. This thesis will focus on a derivation-based approach to noncommutative geometry using the framework of real calculi, which is a rather direct approach to the subject. Due to their direct nature, real calculi are useful when studying classical concepts in Riemannian geometry and how they may be generalized to a noncommutative setting.This thesis aims to shed light on algebraic aspects of real calculi by introducing a concept of morphisms of real calculi, which enables the study of real calculi on a structural level. In particular, real calculi over matrix algebras are discussed both from an algebraic and a geometric perspective.Morphisms are also interpreted geometrically, giving a way to develop a noncommutative theory of embeddings. As an example, the noncommutative torus is minimally embedded into the noncommutative 3-sphere.
41.	Tiger Norkvist, Axel, 1994- (författare) The Noncommutative Geometry of Real Calculi 2023 Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract Noncommutative geometry extends the traditional connections between algebra and geometry beyond the realm of commutative algebras, allowing for a broader exploration of geometric concepts in noncommutative settings. The geometric perspective facilitates the study and understanding of various mathematical structures, including operator algebras, quantum groups, and noncommutative spaces. Since its inception, noncommutative geometry has experienced remarkable growth, attracting mathematicians from diverse backgrounds who seek to delve into the geometric aspects of noncommutative structures. Through this lens, groundbreaking discoveries have deepened our understanding of fundamental mathematical principles and opened up new avenues of research. This ongoing exploration not only enriches our mathematical knowledge but also finds practical applications in theoretical physics, quantum field theory, and interdisciplinary fields.The primary focus of this thesis is to offer valuable insights into the derivation-based approach of real calculi, which employs modules over an algebra as an algebraic analogy for vector bundles over differential manifolds. An overarching goal is to give noncommutative counterparts of classical geometric concepts, with a specific emphasis being placed on a noncommutative adaptation of the Levi-Civita connection in (pseudo-)Riemannian geometry. An investigation into the existence of a Levi-Civita connection is conducted in the context of general projective modules, and in cases where it exists a theory of embeddings is developed and used to give a minimal embedding of the noncommutative torus into the noncommutative 3-sphere. The thesis also establishes the concept of morphisms of real calculi, which plays a crucial role in examining the relationship between projective modules and specific free modules in this framework. Moreover, the thesis provides an in-depth examination of matrix algebras, utilizing them as illustrative examples to showcase the process of determining isomorphism classes of real calculi in various scenarios and presenting classes of examples where a Levi-Civita connection does not exist.

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