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Publications (10 of 20) Show all publications
Arnlind, J., Björn, A. & Björn, J. (2016). An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces. Nonlinear Analysis, 134, 70-104
Open this publication in new window or tab >>An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces
2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 134, p. 70-104Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
PERGAMON-ELSEVIER SCIENCE LTD, 2016
Keywords
Dirichlet problem; First eigenvalue; Generalized Sobolev space; Gradient relation; Lattice; Metric space; Noncommutative function; Obstacle problem; Operator-valued function; Partial order; Poincare set; Rayleigh quotient; Rellich-Kondrachov cone; Trace class ideal; Variational problem
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-126128 (URN)10.1016/j.na.2015.12.010 (DOI)000370489300004 ()
Note

Funding Agencies|Swedish Research Council

Available from: 2016-03-15 Created: 2016-03-15 Last updated: 2017-11-30
Arnlind, J., Choe, J. & Hoppe, J. (2016). Noncommutative Minimal Surfaces. Letters in Mathematical Physics, 106(8), 1109-1129
Open this publication in new window or tab >>Noncommutative Minimal Surfaces
2016 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 106, no 8, p. 1109-1129Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
SPRINGER, 2016
Keywords
noncommutative minimal surfaces; noncommutative Weierstrass representation; Weyl algebra; noncommutative catenoid; noncommutative Enneper surface
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-130372 (URN)10.1007/s11005-016-0861-7 (DOI)000379609000005 ()
Available from: 2016-08-15 Created: 2016-08-05 Last updated: 2017-11-28
Arnlind, J. (2014). Curvature and geometric modules of noncommutative spheres and tori. Journal of Mathematical Physics, 55(4), 041705
Open this publication in new window or tab >>Curvature and geometric modules of noncommutative spheres and tori
2014 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 55, no 4, p. 041705-Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2014
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-107859 (URN)10.1063/1.4871175 (DOI)000336084100007 ()
Available from: 2014-06-23 Created: 2014-06-23 Last updated: 2017-12-05Bibliographically approved
Arnlind, J. & Huisken, G. (2014). Pseudo-Riemannian Geometry in Terms of Multi-Linear Brackets. Letters in Mathematical Physics, 104(12), 1507-1521
Open this publication in new window or tab >>Pseudo-Riemannian Geometry in Terms of Multi-Linear Brackets
2014 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 104, no 12, p. 1507-1521Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Springer Verlag (Germany), 2014
Keywords
Poisson bracket; Nambu bracket; Kahler manifold; Riemannian geometry; submanifolds; multi-linear brackets
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-112804 (URN)10.1007/s11005-014-0723-0 (DOI)000344743400002 ()
Available from: 2015-01-08 Created: 2014-12-17 Last updated: 2017-12-05
Arnlind, J., Kitouni, A., Makhlouf, A. & Silvestrov, S. (2014). Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras. In: ALGEBRA, GEOMETRY AND MATHEMATICAL PHYSICS (AGMP): . Paper presented at Conference on Algebra, Geometry and Mathematical Physics (AGMP) (pp. 123-144). SPRINGER, 85
Open this publication in new window or tab >>Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras
2014 (English)In: ALGEBRA, GEOMETRY AND MATHEMATICAL PHYSICS (AGMP), SPRINGER , 2014, Vol. 85, p. 123-144Conference paper, Published 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.

Place, publisher, year, edition, pages
SPRINGER, 2014
Series
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 85
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-114264 (URN)10.1007/978-3-642-55361-5_9 (DOI)000347610400009 ()978-3-642-55361-5; 978-3-642-55360-8 (ISBN)
Conference
Conference on Algebra, Geometry and Mathematical Physics (AGMP)
Available from: 2015-02-16 Created: 2015-02-16 Last updated: 2015-02-16
Arnlind, J. & Hoppe, J. (2013). The world as quantized minimal surfaces. Physics Letters B, 723(4-5), 397-400
Open this publication in new window or tab >>The world as quantized minimal surfaces
2013 (English)In: Physics Letters B, ISSN 0370-2693, E-ISSN 1873-2445, Vol. 723, no 4-5, p. 397-400Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2013
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-96428 (URN)10.1016/j.physletb.2013.05.022 (DOI)000320745400020 ()
Note

Funding Agencies|Sogang University||

Available from: 2013-08-20 Created: 2013-08-19 Last updated: 2017-12-06
Arnlind, J. & Grosse, H. (2012). Deformed noncommutative tori. Journal of Mathematical Physics, 53(7), 073505
Open this publication in new window or tab >>Deformed noncommutative tori
2012 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 53, no 7, p. 073505-Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2012
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-82075 (URN)10.1063/1.4732099 (DOI)000307609900030 ()
Available from: 2012-09-28 Created: 2012-09-28 Last updated: 2017-12-07
Arnlind, J., Hoppe, J. & Huisken, G. (2012). Multi-linear Formulation of Differential Geometry and Matris Regularizations. Journal of differential geometry, 91(1), 1-39
Open this publication in new window or tab >>Multi-linear Formulation of Differential Geometry and Matris Regularizations
2012 (English)In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 91, no 1, p. 1-39Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
International Press, 2012
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-81838 (URN)000308046900001 ()
Available from: 2012-09-25 Created: 2012-09-24 Last updated: 2017-12-07
Arnlind, J., Makhlouf, A. & Silvestrov, S. (2011). Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras. Journal of Mathematical Physics, 52, Article ID 123502.
Open this publication in new window or tab >>Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras
2011 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, article id 123502Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2011
National Category
Mathematics Mathematical Analysis Algebra and Logic
Identifiers
urn:nbn:se:liu:diva-122351 (URN)10.1063/1.3653197 (DOI)
Available from: 2015-10-29 Created: 2015-10-29 Last updated: 2017-12-01
Arnlind, J. & Hoppe, J. (2010). Discrete Minimal Surface Algebras. SIGMA. Symmetry, Integrability and Geometry, 6(042)
Open this publication in new window or tab >>Discrete Minimal Surface Algebras
2010 (English)In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 6, no 042, p. -18Article in journal (Refereed) Published
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.

Keywords
noncommutative surface; minimal surface; discrete Laplace operator; graph representation; matrix regularization; membrane theory; Yang-Mills algebra.
National Category
Mathematics Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-122352 (URN)10.3842/SIGMA.2010.042 (DOI)
Available from: 2015-10-29 Created: 2015-10-29 Last updated: 2017-12-01
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-8727-2169

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