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Joakim Arnlind

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University of Zagreb

Aalto University

Dr. J. M. Ashfaque (MInstP)

Independent Researcher

New Mexico State University

The University Of Jordan

Kansas State University

Yildiz Technical University

Tomasz Szemberg

University of the National Education Commission in Krakow

Vesna Manojlović

University of Belgrade

Universidad Autónoma de Madrid

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Papers by Joakim Arnlind

Projections, modules and connections for the noncommutative cylinder

Advances in Theoretical and Mathematical Physics

Low Dimensional Matrix Representations for Noncommutative Surfaces of Arbitrary Genus

Mathematical Physics, Analysis and Geometry

In this note, we initiate a study of the finite-dimensional representation theory of a class of a... more 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 $\mathbb {R}^{3}$ ℝ 3 .

A noncommutative catenoid

Letters in Mathematical Physics

A noncommutative algebra corresponding to the classical catenoid is introduced together with a di... more 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.

Kähler–Poisson algebras

Journal of Geometry and Physics

Riemannian curvature of the noncommutative 3-sphere

Journal of Noncommutative Geometry

Multi-linear Formulation of Differential Geometry and Matris Regularizations

Journal of Differential Geometry

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be f... more 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 Weingarten's 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 a large class of explicit examples is 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.

On the Chern–Gauss–Bonnet theorem for the noncommutative 4-sphere

Journal of Geometry and Physics, 2017

We construct a differential calculus over the noncommutative 4sphere in the framework of pseudo-R... more We construct a differential calculus over the noncommutative 4sphere 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 Gauss-Chern-Bonnet type theorem for the noncommutative 4-sphere. 2010 Mathematics Subject Classification. 46L87.

Noncommutative Minimal Surfaces

Letters in Mathematical Physics, 2016

We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct the... more We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass-representation. Given the growing interest in noncommutative spaces, and zero-mean-curvature surfaces having been known for more than 250 years, it is rather astonishing that a general theory of noncommutative minimal surfaces seems to be lacking. Our note is a modest attempt to fill this gap.

Discrete curvature and the Gauss-Bonnet theorem

Eprint Arxiv 1001 2223, Jan 13, 2010

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics... more 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 provide a large class of explicit examples illustrating the new notions.

Noncommutative Riemann Surfaces

We introduce C-Algebras of compact Riemann surfaces Σ as non-commutative analogues of the Poisson... more 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, nicely interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.

More Membrane Matrix Model Solutions - and Minimal Surfaces in S^7

New solutions to the classical equations of motion of a bosonic matrix-membrane are given. Their ... more

On the geometry of K

We prove that the Riemannian geometry of almost Kähler manifolds can be expressed in terms of the... more We prove that the Riemannian geometry of almost Kähler manifolds can be expressed in terms of the Poisson algebra of smooth functions on the manifold. Subsequently, Kähler-Poisson algebras are introduced, and it is shown that a corresponding purely algebraic theory of geometry and curvature can be developed. As an illustration of the new concepts we give an algebraic proof of the statement that a bound on the (algebraic) Ricci curvature induces a bound on the eigenvalues of the (algebraic) Laplace operator, in analogy with the well-known theorem in Riemannian geometry. As the correspondence between Poisson brackets of smooth functions and commutators of operators lies at the heart of quantization, a purely Poisson algebraic proof of, for instance, such a "Gap Theorem", might lead to an understanding of spectral properties in a corresponding quantum mechanical system.

On the classical geometry of embedded manifolds in terms of Nambu brackets

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be f... more We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of a multi-linear algebraic structure on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.

Eigenvalue Dynamics, Follytons and Largen Limits of Matrices

NATO Science Series II: Mathematics, Physics and Chemistry, 2006

ABSTRACT How do the eigenvalues of a “free” hermitian N × N matrix X(t) evolve in time? The answe... more 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.

Graph techniques for matrix equations and eigenvalue dynamics

Classical Solutions in the BMN Matrix Model

Several reductions of the bosonic BMN matrix model equations to ordinary point particle Hamiltoni... more Several reductions of the bosonic BMN matrix model equations to ordinary point particle Hamiltonian dynamics in the plane (or R^3) are given - as well as a few explicit solutions (some of which, as N->infinity, correspond to membranes rotating with constant angular velocity, others to higher dimensional objects).

Affine transformation crossed product type algebras and noncommutative surfaces

Contemporary Mathematics, 2009

Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras

Springer Proceedings in Mathematics & Statistics, 2014

The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and i... more The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced 3-Lie algebras. 1 Introduction Lie algebras have held a very important place in mathematics and physics for a long time. Ternary Lie algebras appeared first in Nambu's generalization of Hamiltonian mechanics [11] which uses a generalization of Poisson algebra with a ternary bracket. The algebraic formulation is due to Takhtajan. The structure of n-Lie algebra was studied by Filippov [8] and Kasymov [10]. The Lie algebra cohomology complex is well known under the name of Chevalley-Eilenberg cohomology complex. The cohomology of n-Lie algebras was first introduced by Takhtajan [13] in its simplest form, later a complex adapted to the study of formal deformations was introduced by Gautheron [9], then reformulated by Daletskii and Takhtajan [5] using the notion of base Leibniz algebra of a n-Lie algebra.

Pseudo-Riemannian Geometry in Terms of Multi-Linear Brackets

Letters in Mathematical Physics, 2014

We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher ... more

Discrete curvature and the Gauss-Bonnet theorem

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics... more 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 provide a large class of explicit examples illustrating the new notions.

Projections, modules and connections for the noncommutative cylinder

Advances in Theoretical and Mathematical Physics

Low Dimensional Matrix Representations for Noncommutative Surfaces of Arbitrary Genus

Mathematical Physics, Analysis and Geometry

In this note, we initiate a study of the finite-dimensional representation theory of a class of a... more 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 $\mathbb {R}^{3}$ ℝ 3 .

A noncommutative catenoid

Letters in Mathematical Physics

A noncommutative algebra corresponding to the classical catenoid is introduced together with a di... more 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.

Kähler–Poisson algebras

Journal of Geometry and Physics

Riemannian curvature of the noncommutative 3-sphere

Journal of Noncommutative Geometry

Multi-linear Formulation of Differential Geometry and Matris Regularizations

Journal of Differential Geometry

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be f... more 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 Weingarten's 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 a large class of explicit examples is 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.

On the Chern–Gauss–Bonnet theorem for the noncommutative 4-sphere

Journal of Geometry and Physics, 2017

We construct a differential calculus over the noncommutative 4sphere in the framework of pseudo-R... more We construct a differential calculus over the noncommutative 4sphere 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 Gauss-Chern-Bonnet type theorem for the noncommutative 4-sphere. 2010 Mathematics Subject Classification. 46L87.

Noncommutative Minimal Surfaces

Letters in Mathematical Physics, 2016

We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct the... more We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass-representation. Given the growing interest in noncommutative spaces, and zero-mean-curvature surfaces having been known for more than 250 years, it is rather astonishing that a general theory of noncommutative minimal surfaces seems to be lacking. Our note is a modest attempt to fill this gap.

Discrete curvature and the Gauss-Bonnet theorem

Eprint Arxiv 1001 2223, Jan 13, 2010

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics... more 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 provide a large class of explicit examples illustrating the new notions.

Noncommutative Riemann Surfaces

We introduce C-Algebras of compact Riemann surfaces Σ as non-commutative analogues of the Poisson... more 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, nicely interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.

More Membrane Matrix Model Solutions - and Minimal Surfaces in S^7

New solutions to the classical equations of motion of a bosonic matrix-membrane are given. Their ... more

On the geometry of K

We prove that the Riemannian geometry of almost Kähler manifolds can be expressed in terms of the... more We prove that the Riemannian geometry of almost Kähler manifolds can be expressed in terms of the Poisson algebra of smooth functions on the manifold. Subsequently, Kähler-Poisson algebras are introduced, and it is shown that a corresponding purely algebraic theory of geometry and curvature can be developed. As an illustration of the new concepts we give an algebraic proof of the statement that a bound on the (algebraic) Ricci curvature induces a bound on the eigenvalues of the (algebraic) Laplace operator, in analogy with the well-known theorem in Riemannian geometry. As the correspondence between Poisson brackets of smooth functions and commutators of operators lies at the heart of quantization, a purely Poisson algebraic proof of, for instance, such a "Gap Theorem", might lead to an understanding of spectral properties in a corresponding quantum mechanical system.

On the classical geometry of embedded manifolds in terms of Nambu brackets

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be f... more We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of a multi-linear algebraic structure on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.

Eigenvalue Dynamics, Follytons and Largen Limits of Matrices

NATO Science Series II: Mathematics, Physics and Chemistry, 2006

ABSTRACT How do the eigenvalues of a “free” hermitian N × N matrix X(t) evolve in time? The answe... more 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.

Graph techniques for matrix equations and eigenvalue dynamics

Classical Solutions in the BMN Matrix Model

Several reductions of the bosonic BMN matrix model equations to ordinary point particle Hamiltoni... more Several reductions of the bosonic BMN matrix model equations to ordinary point particle Hamiltonian dynamics in the plane (or R^3) are given - as well as a few explicit solutions (some of which, as N->infinity, correspond to membranes rotating with constant angular velocity, others to higher dimensional objects).

Affine transformation crossed product type algebras and noncommutative surfaces

Contemporary Mathematics, 2009

Structure and Cohomology of 3-Lie Algebras Induced by Lie Algebras

Springer Proceedings in Mathematics & Statistics, 2014

The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and i... more The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced 3-Lie algebras. 1 Introduction Lie algebras have held a very important place in mathematics and physics for a long time. Ternary Lie algebras appeared first in Nambu's generalization of Hamiltonian mechanics [11] which uses a generalization of Poisson algebra with a ternary bracket. The algebraic formulation is due to Takhtajan. The structure of n-Lie algebra was studied by Filippov [8] and Kasymov [10]. The Lie algebra cohomology complex is well known under the name of Chevalley-Eilenberg cohomology complex. The cohomology of n-Lie algebras was first introduced by Takhtajan [13] in its simplest form, later a complex adapted to the study of formal deformations was introduced by Gautheron [9], then reformulated by Daletskii and Takhtajan [5] using the notion of base Leibniz algebra of a n-Lie algebra.

Pseudo-Riemannian Geometry in Terms of Multi-Linear Brackets

Letters in Mathematical Physics, 2014

We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher ... more

Discrete curvature and the Gauss-Bonnet theorem

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics... more 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 provide a large class of explicit examples illustrating the new notions.