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  • 1.
    Arnlind, Joakim
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Curvature and geometric modules of noncommutative spheres and tori2014In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 55, no 4, p. 041705-Article in journal (Refereed)
    Abstract [en]

    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.

  • 2.
    Arnlind, Joakim
    Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden .
    Representation theory of C  -algebras for a higher-order class of spheres and tori2008In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 49, p. 053502-1-053502-13, article id 053502Article in journal (Refereed)
    Abstract [en]

    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 “Hénon algebras,” for which the dynamical map is a generalized Hénon map, and give an example where irreducible representations of all dimensions exist.

  • 3.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces2016In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 134, p. 70-104Article in journal (Refereed)
    Abstract [en]

    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.

  • 4.
    Arnlind, Joakim
    et al.
    Institut des Hautes Études Scientifiques, Le Bois-Marie 35, route de Chartres, F-91440, Bures-sur-Yvette, France och Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476, Golm, Germany .
    Bordemann, Martin
    Laboratoire de MIA, 4, rue des Frères Lumière, Université de Haute-Alsace, F-68093, Mulhouse, France .
    Hofer, Laurent
    Université du Luxembourg, FSTC 162a, avenue de la Faïencerie, L-1511, Luxembourg City, Luxembourg .
    Hoppe, Jens
    Department of Mathematics, KTH, S-10044, Stockholm, Sweden .
    Shimada, Hidehiko
    Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476, Golm, Germany .
    Noncommutative Riemann Surfaces by Embeddings in R32009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 288, p. 403-429Article in journal (Refereed)
    Abstract [en]

    We introduce C-Algebras of compact Riemann surfaces Σ  as non-commutative analogues of the Poisson algebra of smooth functions on Σ  . Representations of these algebras give 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

  • 5.
    Arnlind, Joakim
    et al.
    Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden.
    Bordemann, Martin
    Laboratoire de MIA, Université de Haute-Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France.
    Hofera, Laurent
    Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden och Laboratoire de MIA, Université de Haute-Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France.
    Hoppe, Jens
    Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden.
    Shimada, Hidehiko
    Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1 D-14476 Golm., Germany.
    Fuzzy Riemann surfaces2009In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, Vol. 2009, no 024, p. 1-17Article in journal (Refereed)
    Abstract [en]

    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.

  • 6.
    Arnlind, Joakim
    et al.
    Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden .
    Bordemann, Martin
    Laboratoire de MIA, 4 rue des Frères Lumière, Univ. deHaute-Alsace, 68093, Mulhouse, France .
    Hoppe, Jens
    Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden .
    Lee, Choonkyu
    Department of Physics and Center for Theoretical Physics, Seoul National University, 151-747, Seoul, South Korea .
    Goldfish Geodesics and Hamiltonian Reduction of Matrix Dynamics2008In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 84, p. 89-98Article in journal (Refereed)
    Abstract [en]

    We describe the Hamiltonian reduction of a time-dependent real-symmetric N×N 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.

  • 7.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Choe, Jaigyoung
    Korea Institute Adv Study, South Korea.
    Hoppe, Jens
    Royal Institute Technology, Sweden.
    Noncommutative Minimal Surfaces2016In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 106, no 8, p. 1109-1129Article in journal (Refereed)
    Abstract [en]

    We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass representation.

  • 8.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Grosse, Harald
    Mathematical Physics, Austria .
    Deformed noncommutative tori2012In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 53, no 7, p. 073505-Article in journal (Refereed)
    Abstract [en]

    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.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Holm, Christoffer
    Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.
    A noncommutative catenoid2018In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 108, no 7, p. 1601-1622Article in journal (Refereed)
    Abstract [en]

    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.

  • 10.
    Arnlind, Joakim
    et al.
    Institut des Hautes Études Scientif iques, Le Bois-Marie, 35, Route de Chartres, 91440 Bures-sur-Yvette, France.
    Hoppe, Jens
    Eidgenössische Technische Hochschule, 8093 Zürich, Switzerland (on leave of absence from Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden).
    Discrete Minimal Surface Algebras2010In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 6, no 042, p. -18Article in journal (Refereed)
    Abstract [en]

    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 sln (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.

  • 11.
    Arnlind, Joakim
    et al.
    Department of Mathematics, Royal Institute of Technology, Stockholm, 100 44, Sweden .
    Hoppe, Jens
    Department of Mathematics, Royal Institute of Technology, Stockholm, 100 44, Sweden .
    EIGENVALUE DYNAMICS, FOLLYTONS AND LARGEN LIMITS OF MATRICES2006In: Applications of Random Matrices in Physics / [ed] Édouard Brézin, Vladimir Kazakov, Didina Serban, Paul Wiegmann, Anton Zabrodin, Springer, 2006, 211, p. 89-94Chapter in book (Refereed)
    Abstract [en]

    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.
    Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden.
    Hoppe, Jens
    Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden.
    Eigenvalue-Dynamics off the Calogero–Moser System2004In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 68, p. 121-129Article in journal (Refereed)
    Abstract [en]

    By finding N(N− 1)/2 suitable conserved quantities, free motions of real symmetric N×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),Xd(0)] has to be purely imaginary, of rank one.

  • 13.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Hoppe, Jens
    Sogang University, South Korea .
    The world as quantized minimal surfaces2013In: Physics Letters B, ISSN 0370-2693, E-ISSN 1873-2445, Vol. 723, no 4-5, p. 397-400Article in journal (Refereed)
    Abstract [en]

    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.

  • 14.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Hoppe, Jens
    Sogang University, South Korea .
    Huisken, Gerhard
    Max Planck Institute for Gravitational Physics, Germany .
    Multi-linear Formulation of Differential Geometry and Matris Regularizations2012In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 91, no 1, p. 1-39Article in journal (Refereed)
    Abstract [en]

    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.

  • 15.
    Arnlind, Joakim
    et al.
    Department of Mathematics, Royal Institute of Technology, 10044 Stockholm, Sweden.
    Hoppe, Jens
    Department of Mathematics, Royal Institute of Technology, 10044 Stockholm, Sweden.
    Theisen, Stefan
    Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Germany.
    Spinning membranes2004In: Physics Letters B, ISSN 0370-2693, E-ISSN 1873-2445, Vol. 599, no 1-2, p. 118-128Article in journal (Refereed)
    Abstract [en]

    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 AdSp×SqAdSp×Sq, as well as in flat space–time. Nontrivial reductions of the BMN matrix model equations are also given.

  • 16.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Huisken, Gerhard
    University of Tubingen, Germany.
    Pseudo-Riemannian Geometry in Terms of Multi-Linear Brackets2014In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 104, no 12, p. 1507-1521Article in journal (Refereed)
    Abstract [en]

    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.

  • 17.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Kitouni, Abdennour
    Université de Haute-Alsace, France.
    Makhlouf, Abdenacer
    Université de Haute-Alsace, France.
    Silvestrov, Sergei
    Mälardalens Högskola, Västerås, Sweden.
    Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras2014In: ALGEBRA, GEOMETRY AND MATHEMATICAL PHYSICS (AGMP), SPRINGER , 2014, Vol. 85, p. 123-144Conference paper (Refereed)
    Abstract [en]

    The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced 3-Lie algebras.

  • 18.
    Arnlind, Joakim
    et al.
    Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
    Makhlouf, Abdenacer
    Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications, 4, rue des Frères Lumière F-68093 Mulhouse, France .
    Silvestrov, Sergei
    Mälardalen University, Division of Applied Mathematics, The School of Education, Culture and Communication, Box 883, 721 23 Västerås, Sweden och Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
    Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras2011In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, article id 123502Article in journal (Refereed)
    Abstract [en]

    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.

  • 19.
    Arnlind, Joakim
    et al.
    Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
    Makhlouf, Abdenacer
    Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France .
    Silvestrov, Sergei
    Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden .
    Ternary Hom–Nambu–Lie algebras induced by Hom–Lie algebras2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, p. 043515-1-043515-11, article id 43515Article in journal (Refereed)
    Abstract [en]

    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.

  • 20.
    Arnlind, Joakim
    et al.
    Mathematical Physics, Royal Institute of Technology, SE-106 91, Stockholm, Sweden .
    Mickelsson, Jouko
    Mathematical Physics, Royal Institute of Technology, SE-106 91, Stockholm, Sweden .
    Trace Extensions, Determinant Bundles, and Gauge Group Cocycles2002In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 62, no 2, p. 101-110Article in journal (Refereed)
    Abstract [en]

    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.

  • 21.
    Arnlind, Joakim
    et al.
    Albert Einstein Institute, Golm, Germany..
    Silvestrov, Sergei
    Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
    Affine transformation crossed product type algebras and noncommutative surfaces2009In: Operator structures and dynamical systems :: July 21-25 2008, Lorentz Center, Leiden, the Netherlands, satellite conference of the fifth European Congress of Mathematics, American Mathematical Society (AMS), 2009, 503, p. 1-25Chapter in book (Refereed)
    Abstract [en]

    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.

  • 22.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Wilson, Mitsuru
    University of Western Ontario, Canada.
    On the Chern-Gauss-Bonnet theorem for the noncommutative 4-sphere2017In: JOURNAL OF GEOMETRY AND PHYSICS, ISSN 0393-0440, Vol. 111, p. 126-141Article in journal (Refereed)
    Abstract [en]

    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.

  • 23.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Wilson, Mitsuru
    University of Western Ontario, Canada.
    Riemannian curvature of the noncommutative 3-sphere2017In: Journal of Noncommutative Geometry, ISSN 1661-6952, E-ISSN 1661-6960, Vol. 11, no 2, p. 507-536Article in journal (Refereed)
    Abstract [en]

    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.

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