scholarly journals Cyclic homology of cleft extensions of algebras

2018 ◽  
Vol 17 (05) ◽  
pp. 1850091
Author(s):  
Jorge A. Guccione ◽  
Juan J. Guccione ◽  
Christian Valqui
Keyword(s):  
Commutative Algebra ◽  
Cyclic Homology ◽  
Mixed Complex ◽  
Harmonic Decomposition ◽  
The One ◽  
Cleft Extension ◽  
Cleft Extensions

Let [Formula: see text] be a commutative algebra with [Formula: see text] and let [Formula: see text] be a cleft extension of [Formula: see text]. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of [Formula: see text] relative to [Formula: see text]. This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra.

Cleft extensions and Galois extensions for Hom-associative algebras

2016 ◽  
Vol 27 (03) ◽  
pp. 1650025 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Associative Algebra ◽  
Galois Extension ◽  
Classical Case ◽  
Normal Basis ◽  
Associative Algebras ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions

In this paper, we consider Hom-(co)modules associated to a Hom-(co)associative algebra and define the notion of Hom-triple. We introduce the definitions of cleft extension and Galois extension with normal basis in this setting and we show that, as in the classical case, these notions are equivalent in the Hom setting.

Cleft extensions for quasi-entwining structures

2018 ◽  
Vol 68 (2) ◽  
pp. 339-352
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cleft Extension ◽  
Cleft Extensions ◽  
Necessary And Sufficient

Abstract In this paper we introduce the notions of quasi-entwining structure and cleft extension for a quasi-entwining structure. We prove that if (A, C, ψ) is a quasi-entwining structure and the associated extension to the submagma of coinvariants AC is cleft, there exists an isomorphism ωA between AC ⊗ C and A. Moreover, we define two unital but not necessarily associative products on AC ⊗ C. For these structures we obtain the necessary and sufficient conditions to assure that ωA is a magma isomorphism, giving some examples fulfilling these conditions.

On equivalence of two approaches in index theory

2008 ◽  
Vol 2 (3) ◽  
pp. 445-452
Author(s):  
V. Manuilov
Keyword(s):  
Commutative Algebra ◽  
Compact Manifold ◽  
Pseudodifferential Operators ◽  
Index Theory ◽  
Index Theorem ◽  
Continuous Functions ◽  
Zero Section ◽  
Order Zero ◽  
The One

AbstractThe algebra Ψ(M) of order zero pseudodifferential operators on a compact manifoldMdefines a well-knownC*-extension of the algebraC(S*M) of continuous functions on the cospherical bundleS*M⊂T*Mby the algebra К of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphismTfromC0(T*M) to К, which plays the role of a deformation for the commutative algebraC0(T*M). Similar constructions exist also for operators and symbols with coefficients in aC*-algebra. Recently we have shown that the image of the above extension under the Connes–Higson construction isTand that this extension can be reconstructed out ofT. That is why the classical approach to the index theory coincides with the one based on asymptotic homomorphisms. But the image of the above extension is defined only outside the zero section ofT*(M), so it may seem that the information encoded in the extension is not the same as that in the asymptotic homomorphism. We show that this is not the case.

DYNAMICAL SYSTEMS IN GRAPH C*-ALGEBRAS

2005 ◽  
Vol 16 (09) ◽  
pp. 999-1015 ◽  
Author(s):  
JA A JEONG ◽  
GI HYUN PARK
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Commutative Algebra ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Locally Finite ◽  
C Algebra ◽  
C Algebras ◽  
Shift Space ◽  
The One

Let E be a row finite directed graph with no sinks and (XE, σE) the one-sided edge shift space. Then the graph C*-algebra C*(E) contains the commutative algebra C0(XE). Moreover if E is locally finite so that the canonical completely positive map ϕE on C*(E) is well-defined, ϕE|C0(XE) coincides with the *-homomorphism [Formula: see text]. In this paper we first show that if two edge shift spaces (XE, σE) and (XF, σF) are topologically conjugate, there is an isomorphism of C*(E) onto C*(F), and if the graphs are locally finite the isomorphism transforms ϕE|C0(XE) onto ϕF|C0(XF), which has been known for Cuntz–Krieger algebras. Let ht(ϕE) be Voiculescu–Brown topological entropy of ϕE. In case E is finite, it is well-known that the values ht(ϕE), [Formula: see text], hl(E) and hb(E) all coincide, where [Formula: see text] is the AF core of C*(E) and hl(E), hb(E) are the loop, block entropies of E respectively. If E is irreducible and infinite, [Formula: see text] has been known recently, and here we show that [Formula: see text], where Et is the transposed graph of E. Also some dynamical systems related with AF subalgebras [Formula: see text] of [Formula: see text] are examined to prove that [Formula: see text] for each vertex v.

scholarly journals Split Lie–Rinehart algebras

2020 ◽  
pp. 2150164
Author(s):  
Helena Albuquerque ◽  
Elisabete Barreiro ◽  
A. J. Calderón ◽  
José M. Sánchez
Keyword(s):  
Lie Algebras ◽  
Commutative Algebra ◽  
Natural Extension ◽  
Nonzero Ideal ◽  
Direct Sums ◽  
Mild Conditions ◽  
The Family ◽  
The One

We introduce the class of split Lie–Rinehart algebras as the natural extension of the one of split Lie algebras. We show that if [Formula: see text] is a tight split Lie–Rinehart algebra over an associative and commutative algebra [Formula: see text] then [Formula: see text] and [Formula: see text] decompose as the orthogonal direct sums [Formula: see text] and [Formula: see text], where any [Formula: see text] is a nonzero ideal of [Formula: see text], any [Formula: see text] is a nonzero ideal of [Formula: see text], and both decompositions satisfy that for any [Formula: see text], there exists a unique [Formula: see text] such that [Formula: see text]. Furthermore, any [Formula: see text] is a split Lie–Rinehart algebra over [Formula: see text]. Also, under mild conditions, it is shown that the above decompositions of [Formula: see text] and [Formula: see text] are by means of the family of their, respective, simple ideals.

scholarly journals Weak quasi-entwining structures

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6229-6252
Author(s):  
Álvarez Alonso ◽  
Vilaboa Fernádez ◽  
González Rodríguez
Keyword(s):  
Galois Extension ◽  
Hopf Algebras ◽  
Normal Basis ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions ◽  
Weak Hopf Algebras

In this paper we introduce the notion of weak quasi-entwining structure as a generalization of quasi-entwining structures and weak entwining structures. Also, we formulate the notions of weak cleft extension, weak Galois extension, and weak Galois extension with normal basis associated to a weak quasientwining structure. Moreover, we prove that, under some suitable conditions, there exists an equivalence between weak Galois extensions with normal basis and weak cleft extensions. As particular instances, we recover some results previously proved for Hopf quasigroups, weak Hopf quasigroups and weak Hopf algebras.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

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