scholarly journals A residue formula for the fundamental Hochschild class on the Podleś sphere

2013 ◽  
Vol 12 (2) ◽  
pp. 257-271 ◽  
Author(s):  
Ulrich Krähmer ◽  
Elmar Wagner
Keyword(s):  
Cohomology Class ◽  
Hochschild Cohomology ◽  
Spectral Triple ◽  
Quantum Sphere ◽  
Residue Formula

AbstractThe fundamental Hochschild cohomology class of the standard Podleś quantum sphere is expressed in terms of the spectral triple of Dąabrowski and Sitarz by means of a residue formula.

scholarly journals Missing the point in noncommutative geometry

Synthese ◽  
2021 ◽  
Author(s):  
Nick Huggett ◽  
Fedele Lizzi ◽  
Tushar Menon
Keyword(s):  
Scalar Field ◽  
Noncommutative Geometry ◽  
Smooth Manifold ◽  
Operational Definition ◽  
Spectral Triple ◽  
Fundamental Quantity

AbstractNoncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

scholarly journals The Gerstenhaber structure on the Hochschild cohomology of a class of special biserial algebras

2021 ◽  
Vol 580 ◽  
pp. 264-298
Author(s):  
Joanna Meinel ◽  
Van C. Nguyen ◽  
Bregje Pauwels ◽  
María Julia Redondo ◽  
Andrea Solotar
Keyword(s):  
Hochschild Cohomology ◽  
Special Biserial Algebras

scholarly journals Going from cohomology to Hochschild cohomology

2005 ◽  
Vol 288 (2) ◽  
pp. 263-278 ◽  
Author(s):  
Emil Sköldberg
Keyword(s):  
Hochschild Cohomology

scholarly journals Gerstenhaber brackets on Hochschild cohomology of twisted tensor products

2017 ◽  
Vol 11 (4) ◽  
pp. 1351-1379 ◽  
Author(s):  
Lauren Grimley ◽  
Van Nguyen ◽  
Sarah Witherspoon
Keyword(s):  
Hochschild Cohomology ◽  
Tensor Products

Bundles over Quantum Sphere and Noncommutative Index Theorem

K-Theory ◽  
2000 ◽  
Vol 21 (2) ◽  
pp. 141-150 ◽  
Author(s):  
Piotr M. Hajac
Keyword(s):  
Index Theorem ◽  
Quantum Sphere

scholarly journals On Hochschild Cohomology of Preprojective Algebras, I

1998 ◽  
Vol 205 (2) ◽  
pp. 391-412 ◽  
Author(s):  
Karin Erdmann ◽  
Nicole Snashall
Keyword(s):  
Hochschild Cohomology ◽  
Preprojective Algebras

scholarly journals Algebra endomorphisms and derivations of some localized down-up algebras

2014 ◽  
Vol 14 (03) ◽  
pp. 1550034 ◽  
Author(s):  
Xin Tang
Keyword(s):  
Automorphism Group ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Root Of Unity ◽  
Hochschild Cohomology Group ◽  
Algebra Endomorphism

We study algebra endomorphisms and derivations of some localized down-up algebras A𝕊(r + s, -rs). First, we determine all the algebra endomorphisms of A𝕊(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A𝕊(r + s, -rs) is an algebra automorphism if rmsn = 1 implies m = n = 0. When r = s-1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A𝕊(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A𝕊(r + s, -rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r + s, -rs) is an algebra automorphism in either case. Second, we determine all the derivations of A𝕊(r + s, -rs) and calculate its first degree Hochschild cohomology group.

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