Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension

Arutyunov, Andronick; Mishchenko, А. S.

doi:10.1134/s0001434613090174

Cited by 3 publications

(8 citation statements)

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“…This allows us to enlarge the class of pseudodifferential operators studied in [5] and [6] by supplementing them with nonlocal pseudodifferential operators (pseudodifferential operators with shifts).…”

Section: Application Of the Reduction In The Case Of Nonlocal Pdomentioning

confidence: 99%

“…We need the results obtained earlier in [5] and [6] by the author and Professor A. S. Mishchenko. Let us recall the basic results obtained in these papers.…”

Section: Introductionmentioning

confidence: 99%

“…The pseudodifferential operators acting in the Schwartz space S(R n ), with symbols in the class S m 1 ,m 2 (R n × R n ), are called bigraded operators of generalized order (m 1 , m 2 ). It was shown in [5] and [6] that the operators with such symbols can be identified with pseudodifferential operators acting in the space of smooth sections of a one-dimensional fibration over the torus T 2n of doubled dimension. The space M (R 2n ) is also understood as the functional space of infinitely differentiable functions defined on the Euclidean space R 2n and satisfying the skew-periodicity condition.…”

Section: Introductionmentioning

confidence: 99%

“…The transformation Λ is an isomorphism between the spaces S(R n ) and M (R 2n ) (see [5], [6]). Here and below, we let I n denote the standard n-dimensional unit cube with vertices x i ∈ {0, 1}, i = 1, .…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1 is used in [5] and [6] to define the reduction of pseudodifferential operators over the Schwartz space S(R n ) to pseudodifferential operators over the space M (R 2n ). Let us recall the definition of pseudodifferential operators (see [9, p. 28]).…”

Section: Introductionmentioning

confidence: 99%

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Reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a double-dimensional compact manifold

Arutyunov

2015

Math Notes

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In the present paper, we obtain some Fredholmness criteria and a formula for the index of nonlocal pseudodifferential operators generated by shifts and multiplications by periodic functions and acting in the Schwartz space S(R n ). These results significantly differ from the results known to the author in this field of study, namely, the Fredholmness criteria and the formula for the index of nonlocal pseudodifferential operators acting over the noncompact manifold R n are obtained here for the first time.

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Section: Application Of the Reduction In The Case Of Nonlocal Pdomentioning

confidence: 99%

“…We need the results obtained earlier in [5] and [6] by the author and Professor A. S. Mishchenko. Let us recall the basic results obtained in these papers.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a double-dimensional compact manifold

Arutyunov

2015

Math Notes

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Chapter 3: Pseudo-Differential Operators and Fourier Operators

Demidov

2023

Equations of Mathematical Physics

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Geometric description of the Hochschild cohomology of group algebras

Mishchenko¹

2021

Topology, Geometry, and Dynamics

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There are two approaches to the study of the cohomology of group algebras R [ G ] R[G] , the Eilenberg–MacLane cohomology and the Hochschild cohomology. The Eilenberg–MacLane cohomology gives the classical cohomology of the classifying space B G BG (or the Eilenberg–MacLane complex K ( G , 1 ) K(G,1) ). Note that the space B G BG can be interpreted as a classifying space of the groupoid of the trivial action of the group G G . The Hochschild cohomology is a more general construction, which considers the so-called bimodules of the algebra R [ G ] R[G] and their derivative functors Ext ⁡ ( R [ G ] , M ) \operatorname {Ext}(R[G],M) , for which no geometric interpretation has been known so far. The key point for calculating the Hochschild cohomology H H ∗ ( R [ G ] ) HH^*(R[G]) is the new groupoid G r Gr associated with the adjoint action of the group G G . For this groupoid, the classical cohomology of the corresponding classification space B G r BGr with the finiteness condition for the supports of cochains is isomorphic to the Hochschild cohomology of the algebra R [ G ] R[G] : H H ∗ ( R [ G ] ) ≈ H f ∗ ( B G r ) . \begin{equation*} HH^*(R[G])\approx H^*_f(BGr). \end{equation*} This result represents a fundamental contribution to understanding the geometry of the cohomological properties of group algebras, in particular, understanding the differences between the homology and cohomology of group algebras. The paper is devoted to the motivation of the Hochschild (co)homology group of the group algebra R [ G ] R[G] and its description in terms of the classical (co)homology of the classifying space of the groupoid of the adjoint action of the original group G G under a suitable finiteness assumption on the supports of the cohomology group.

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Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension

Cited by 3 publications

References 3 publications

Reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a double-dimensional compact manifold

Reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a double-dimensional compact manifold

Chapter 3: Pseudo-Differential Operators and Fourier Operators

Geometric description of the Hochschild cohomology of group algebras

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