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3f34a806953bb25e671a2ebd855cbe358f6e0265 | subsection | 4 | 63 | Introduction | By learning how to proceed to find conserved currents within the original general Horndeski framework, one should be able to tackle similarly any of these generalized theories. It is also important to notice that as a special cases of the Horndeski system one can extract any of the so-called f(R) theories or the protot... | {
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c5a6d0d46f4e0b23a766e7921b3911e907ea7e60 | subsection | 5 | 63 | Introduction | VI). Regarding the applications of the KBL formalism in, for example, a cosmological context, we study superpotentials associated with perturbations of metric and scalar fields in Sec. VII. Here, the results for linear perturbations are presented in detail. The final expressions are very lengthy; however, they consider... | {
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cb46f81c08738a206bb53f4d2e3167bbeccba79b | subsection | 6 | 63 | The Horndeski scalar-tensor theory | The Horndeski theory, the most general scalar-tensor theory of gravitation with the second-order field equations, was originally developed in Ref. horndeski and then rederived in Ref. deffayet. The theory with field variables being scalar field \varphi and metric tensor g_{\mu \nu } is given by the Lagrangian\hat{\math... | {
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fe7ab5955c127ba323c779a5772c4efec76acebb | subsection | 7 | 63 | Formulas for arbitrary Lagrangian theory | Methods of obtaining conserved currents from expressions involving general Lagrangians are described abundantly in literature. Our aim is to derive the “covariantized Noether identities” by introducing an auxiliary metric. This metric, considered to be associated with a given background spacetime, provides the tool to ... | {
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6788c4280cdadd2e357e704fc6c9698ee0708389 | subsection | 8 | 63 | Formulas for arbitrary Lagrangian theory | (37), (41), (44), and (47))\partial _{\alpha } \hat{i}^{\alpha } = \partial _{\alpha } \left( \hat{u}^{\alpha }_{\sigma } \xi ^{\sigma } + \hat{m}^{\alpha \tau }_{\sigma } \bar{\nabla }_{\tau }\xi ^{\sigma } + \hat{n}^{\alpha \tau \beta }_{\sigma } \bar{\nabla }_{\tau \beta } \xi ^{\sigma } \right) = 0,which represents... | {
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2fa4406a9f2d095b64dddbb7e68799ced639ed51 | subsection | 9 | 63 | Formulas for arbitrary Lagrangian theory | Q_B \right|^{\alpha }_{\sigma } - \left[ \frac{\partial \hat{\mathcal {L}}}{\partial Q_{B|\alpha }} - \bar{\nabla }_{\beta } \left( \frac{\partial \hat{\mathcal {L}}}{\partial Q_{B|\alpha \beta }} \right) \right] \bar{\nabla }_{\sigma } Q_B - \frac{\partial \hat{\mathcal {L}}}{\partial Q_{B|\beta \alpha }} \bar{\nabla ... | {
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622925e12d5609f59d54e27c3a30adbd8d8614fc | subsection | 10 | 63 | The scalar field | In Lagrangians (REF )–(), we have the first derivatives of scalar field \varphi in quadratic term X = - \frac{1}{2}\partial _{\mu } \varphi \partial ^{\mu } \varphi and the second derivatives in terms \nabla _{\mu \nu } \varphi , \Box \varphi , \mbox{Tr} \, \Pi ^2 and \mbox{Tr} \, \Pi ^3.Since \partial _{\alpha } \varp... | {
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857f8bc17aeedccb726861bb7ca28a70cc2cafe2 | subsection | 11 | 63 | The scalar field | Whenever covariant derivatives instead of partial derivatives are employed like in the quantities \Delta ^{\lambda }_{\mu \nu } (REF ), they are constructed using the auxiliary metric \bar{g}_{\mu \nu } and are associated with a lower case index (like in \bar{\nabla }_{\nu } g_{\rho \mu }). However, because of the pres... | {
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47b555fd847405d1d37a766dfd69e836e2c38d87 | subsection | 12 | 63 | The metric field | The derivatives of metric field g_{\mu \nu } are present in Lagrangians \mathcal {L}_4 and \mathcal {L}_5 [see () and ()] in the form of the Ricci scalar R and the Einstein tensor G_{\mu \nu }. The covariant version of the Riemann tensor R{^{\lambda }}_{\tau \rho \sigma }, which can be found, e.g., in Ref. KBL, isR{^{\... | {
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} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
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c2ea880c5335c878be87e8205b823ab9fd02c03f | subsection | 13 | 63 | The metric field | In this paper, we will use the covariantization in which this whole antisymmetric part is inserted into the second part Q{^{\lambda }}_{\tau \rho \sigma }, leading thus toR{^{\lambda }}_{\tau \rho \sigma } &= \frac{1}{2}g^{\lambda \iota } \left( \bar{\nabla }_{(\rho \tau )} g_{\iota \sigma } - \bar{\nabla }_{(\rho \iot... | {
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1b2fad1ec598af3d10bc504f5a568d3e316f1e9e | subsection | 14 | 63 | Conserved currents formulas in the Horndeski theory | Since in the Horndeski theory there are two different sets of the field variables, Q_B \equiv (\varphi , g_{\mu \nu }), we split the coefficients \hat{u}^{\alpha }_{\sigma }, \hat{m}_{\sigma }^{\alpha \tau } and \hat{n}_{\sigma }^{\alpha \tau \beta } forming the conserved current \hat{i}^{\alpha } (REF ) into gravitati... | {
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ea944d02f9b02a11d213dbcd2c8e305474f6e3e2 | subsection | 15 | 63 | Conserved currents formulas in the Horndeski theory | \varphi \right|^{\tau }_{\sigma } = 0, hence, the formulas (REF ), () and () considerably simplify:\hat{n}^{\alpha \tau \beta }_{\sigma (\varphi )} &= 0, \\
\hat{m}^{\alpha \tau }_{\sigma (\varphi )} &= - \frac{\partial \hat{\mathcal {L}}}{\partial \varphi _{|\tau \alpha }} \bar{\nabla }_{\sigma } \varphi , \\
\hat{u}^... | {
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} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
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28a683756efa86aa10520095ef384c34e493c9a8 | subsection | 16 | 63 | Conserved currents formulas in the Horndeski theory | This leads to the following formulas for the conserved current coefficients:\hat{n}^{\alpha \tau \beta }_{\sigma (g)} &= -g_{\rho \sigma } \left( \frac{\partial \hat{\mathcal {L}}}{\partial g_{\tau \rho |\beta \alpha }} + \frac{\partial \hat{\mathcal {L}}}{\partial g_{\beta \rho |\tau \alpha }} \right), \\
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} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
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3f0c7d8967426626f9ff1fdfe0edf5185b2cc34a | subsection | 17 | 63 | The derivatives of the Horndeski Lagrangian | We give some intermediate results necessary to calculate the conserved current coefficients \hat{u}^{\alpha }_{\sigma }, \hat{m}_{\sigma }^{\alpha \tau }, and \hat{n}_{\sigma }^{\alpha \tau \beta }. Simultaneously, it is rather illustrative to see how the covariantization and differentiation with respect to auxiliary f... | {
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b420c84ab01bcb98a2f45fed992d6d7c15423b8b | subsection | 18 | 63 | Derivatives with respect to the scalar field | The Horndeski Lagrangian consists of several constituents containing derivatives of field \varphi : the kinetic term X, the d'Alambertian \Box \varphi and the trace of powers of matrix \Pi , \mbox{Tr}(\Pi ^k), k = 2, \, 3. For the calculation of the derivatives we need to write these terms manifestly covariant with res... | {
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28984cf7915f669e99a7d9e88966b6b59ee0c123 | subsection | 19 | 63 | Derivatives with respect to the scalar field | Hence,\frac{\partial X}{\partial \varphi _{|\alpha }} &= - g^{\alpha \rho } \nabla _{\rho } \varphi = - \nabla ^{\alpha } \varphi , & \frac{\partial X}{\partial \varphi _{|\alpha \beta }} &= 0.Notice that for raising and lowering indices only the metric field g_{\mu \nu } is always used.The derivatives of \nabla _{\mu ... | {
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003cce7108e685e8ab3595d96baa1220591c595e | subsection | 20 | 63 | Derivatives with respect to the metric field | The metric field g_{\mu \nu } is explicitly present in the Ricci scalar and the Riemann tensor, and its first derivatives are also contained in the second derivatives of the scalar field \varphi , in the expression for the Christoffel symbols difference [see (REF ) and (REF )]. | {
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1a7c7912fb3ee26709afefa52d21384cb9ba5e63 | subsection | 21 | 63 | Derivatives with respect to the metric field | Hence, using\frac{\partial \Delta ^{\lambda }_{\tau \sigma }}{\partial g_{\mu \nu |\alpha }} = \delta ^{\alpha }_{(\tau } \delta ^{(\mu }_{\sigma )} g^{\nu )\lambda } - \frac{1}{2}\delta ^{(\mu }_{(\tau } \delta ^{\nu )}_{\sigma )} g^{\alpha \lambda },we find the following relation:\frac{\partial \varphi _{;\kappa \lam... | {
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997c4bc5c45922f3ade7a502836c063d9578c5f4 | subsection | 22 | 63 | Derivatives with respect to the metric field | REF and REF and the Appendix. | {
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2a204e390cb5ef35317870ddfd4fc100ff2c7a01 | subsection | 23 | 63 | Conserved current coefficients for | Starting from (REF ) and employing the preceding results, we easily obtain the conserved current coefficients \hat{u}^{\alpha }_{\sigma (2)}, \hat{m}_{\sigma (2)}^{\alpha \tau } and \hat{n}_{\sigma (2)}^{\alpha \tau \beta }. First, derivatives with respect to the scalar field derivatives are\frac{\partial \mathcal {L}_... | {
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"hep-th"
] | 2,018 | en | Physics | [
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2386e1045f9642bd038f83cdd303ee8f7ed0db60 | subsection | 24 | 63 | Conserved current coefficients for | These have an obvious impact on the derivative of \mathcal {L}_3 w.r.t. the second partial derivative of the field \varphi , but they also imply the non-vanishing derivatives w.r.t. the first derivatives of the field and metric g_{\mu \nu } [see (REF ) and (REF )].For the scalar part we get\frac{\partial \mathcal {L}_3... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0.021433532238006592,
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-0... |
dec7d08c21e387bc8ac8aeb6514c846beaf3d434 | subsection | 25 | 63 | Conserved current coefficients for | For the derivative of the metric and its determinant, we have\bar{\nabla }_{\alpha } g^{\mu \nu } = - 2 g^{\lambda (\mu } \Delta ^{\nu )}_{\lambda \alpha }, \qquad \bar{\nabla }_{\alpha } g_{\mu \nu } = 2 g_{\lambda (\mu } \Delta ^{\lambda }_{\nu )\alpha };and \bar{\nabla }_{\alpha } g = 2 g \Delta ^{\lambda }_{\lambda... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
0.007806889712810516,
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0.03260120376944542,
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-0.... |
d95fcf13930bd412b43e4da0e8ce01a2da92a493 | subsection | 26 | 63 | Conserved current coefficients for | The non-vanishing derivatives of \mathcal {L}_3 are obtained using (REF ). The result is\frac{\partial \mathcal {L}_3}{\partial g_{\mu \nu |\alpha }} &= G_3 \left( g^{\alpha (\mu } \nabla ^{\nu )} \varphi - \frac{1}{2}g^{\mu \nu } \nabla ^{\alpha } \varphi \right), \\
\frac{\partial \mathcal {L}_3}{\partial g_{\mu \nu ... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.05251024290919304,
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0.00667... |
acd1a9aeda15c95959a565320564a44cf97e765c | subsection | 27 | 63 | Conserved current coefficients for | The calculations for conserved current coefficients will be performed in such a way that the coefficients corresponding to the Einstein-Hilbert Lagrangian will be preserved in the final result.The derivatives of \mathcal {L}_4 with respect to the scalar field derivatives are easily obtained using expressions () and (RE... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.028437761589884758,
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0.01894833706319332,
-0.027339307591319084,
... |
1c9b288f6c9deb5961a12e783c4874d0be590aa1 | subsection | 28 | 63 | Conserved current coefficients for | The resulting conserved current coefficients for scalar part then read\hat{m}^{\alpha \tau }_{\sigma (\varphi )(4)} &= - \partial _X \hat{G}_4 \left[ 2 \Box \varphi \, g^{\tau \alpha } - 2 \nabla ^{\alpha \tau } \varphi \right] \nabla _{\sigma } \varphi , \\
\hat{u}^{\alpha }_{\sigma (\varphi )(4)} =& \, \partial _{X} ... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0.024228082969784737,
-0.027447305619716644,
-0... |
0ebaa64117af129704c2ea0ec8b27b1bfdee0e17 | subsection | 29 | 63 | Conserved current coefficients for | Employing () and (REF ), we obtain the Lagrangian derivatives with respect to the metric field derivatives as follows:\frac{\partial \mathcal {L}_4}{\partial g_{\mu \nu |\alpha }} &= G_4 \frac{\partial R}{\partial g_{\mu \nu |\alpha }} + \partial _X G_4 \left[ g^{\mu \nu } \nabla ^{\alpha } \varphi \, \Box \varphi - 2 ... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.015643630176782608,
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0.014773691073060036,
-0.01753612793982029,
0.0076... |
03c902d3db94ec81e0bccea10988957eb098550b | subsection | 30 | 63 | Conserved current coefficients for | The second part will contribute to the Einstein-Hilbert coefficient \hat{m}^{\alpha \tau }_{\sigma (EH)}, for the first part we need an explicit expression for scalar curvature derivative,\frac{\partial R}{\partial g_{\mu \nu |\alpha \beta }} &= g^{\alpha (\mu } g^{\nu )\beta } - g^{\alpha \beta } g^{\mu \nu }.Using ex... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.03564209118485451,
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0.017866820096969604,
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0.00... |
1de0918197894bcd76638c34997c0f5b9239dbc2 | subsection | 31 | 63 | Conserved current coefficients for | \\
& \quad \, - 2 \Delta ^{\rho }_{\sigma \kappa } \nabla _{\rho } \varphi \nabla ^{\kappa \alpha } \varphi - 2 \Delta ^{\rho }_{\sigma \kappa } \nabla ^{\kappa } \varphi \nabla {_{\rho }}^{\alpha } \varphi + 2 \Delta ^{\rho }_{\rho \sigma } \nabla _{\kappa } \varphi \nabla ^{\kappa \alpha } \varphi - \Delta ^{\alpha }... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.015282372944056988,
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0.031022530049085617,
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... |
429e8a6a90027afd7fab653604733943fae227d2 | subsection | 32 | 63 | Conserved current coefficients for | The derivatives with respect to the scalar field are found rather easily using (REF ), (), (REF ) and ():\frac{\partial \mathcal {L}_5}{\partial \varphi _{|\alpha }} =& \, - G_5 G^{\mu \nu } \Delta ^{\alpha }_{\mu \nu } + \partial _X G_5 \bigg ( - G^{\mu \nu } \nabla ^{\alpha } \varphi \nabla _{\mu \nu } \varphi + \fra... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0.009674793109297752,
-0.06409168988466263,
... |
ccf78d255ad407564d291fc736adaa0506372e41 | subsection | 33 | 63 | Conserved current coefficients for | Also, we use the following relation:\bar{\nabla }_{\beta } G^{\alpha \beta } = - \left( \Delta ^{\alpha }_{\rho \beta } G^{\rho \beta } + \Delta ^{\beta }_{\rho \beta } G^{\alpha \rho } \right).After quite tedious calculations we get the final form of the conserved current coefficient \hat{u}^{\alpha }_{\sigma (\varphi... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0.015223133377730846,
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0.0... |
882a20120e57007329709521207827e7fb73d9e8 | subsection | 34 | 63 | Conserved current coefficients for | \left( \nabla _{\rho \sigma } \varphi + \Delta ^{\kappa }_{\rho \sigma } \nabla _{\kappa } \varphi \right) \left( \frac{1}{2}g^{\alpha \rho } \left( (\Box \varphi )^2 - \mbox{Tr} \Pi ^2 \right) - \Box \varphi \nabla ^{\alpha \rho } \varphi + \nabla ^{\rho \lambda } \varphi \nabla {_{\lambda }}^{\alpha } \varphi \right)... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0.025999968871474266,
0.013015243224799633,
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0.0032232943922281265,
0.0009493459947407246... |
b14ec9ef5ad6e8567da874b6ce02e58d2aa33bfb | subsection | 35 | 63 | Conserved current coefficients for | The derivative of \mathcal {L}_5 with respect to the first derivative of g_{\mu \nu } is worked out using expressions (), (REF ), and ():\frac{\partial \mathcal {L}_5}{\partial g_{\mu \nu |\alpha }} =& \, G_5 \left( \frac{\partial R^{\rho \kappa }}{\partial g_{\mu \nu |\alpha }} \nabla _{\rho \kappa } \varphi - \frac{1... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0.... |
6095bcdbb2506451290b79b34414ad882721f57d | subsection | 36 | 63 | Conserved current coefficients for | Using the derivative of Ricci tensor with respect to the first derivative of the metric,\frac{\partial R_{\tau \sigma }}{\partial g_{\mu \nu |\alpha }} =& \, \frac{1}{2}\Delta ^{\alpha }_{\rho \kappa } \delta ^{(\mu }_{(\tau } \delta ^{\nu )}_{\sigma )} g^{\rho \kappa } - \Delta ^{\alpha }_{\rho (\tau } \delta ^{(\mu }... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0... |
60ffeb6acb9c3705862972b09fefd6d99ade84a3 | subsection | 37 | 63 | Conserved current coefficients for | The differentiated Ricci tensor with respect to the second derivatives of the metric reads\frac{\partial R_{\tau \sigma }}{\partial g_{\mu \nu |\alpha \beta }} =& \, \frac{1}{2}\left( \delta ^{\alpha }_{(\tau } \delta ^{(\mu }_{\sigma )} g^{\nu )\beta } + \delta ^{\beta }_{(\tau } \delta ^{(\mu }_{\sigma )} g^{\nu )\al... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0.009590290486812592,
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... |
dd246d95a6a121a7e1569f88971b776b5a14e2bc | subsection | 38 | 63 | Conserved current coefficients for | \\
& \, \qquad \left. + \nabla _{\rho } \varphi \left( \delta ^{\tau }_{\sigma } \nabla ^{\rho \kappa } \varphi \nabla {_{\kappa }}^{\alpha } \varphi - \frac{1}{2}\delta ^{\alpha }_{\sigma } \nabla ^{\rho \kappa } \varphi \nabla {_{\kappa }}^{\tau } \varphi - \frac{1}{2}g^{\alpha \tau } \nabla ^{\rho \kappa } \varphi \... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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0.010477747768163681,
-0.041117336601018906,
-0.0015... |
c2b085a6a6257376360f46e9c1a1c7071677aa14 | subsection | 39 | 63 | Conserved current coefficients for | \\
& \, \qquad \left. - \frac{1}{2}\nabla ^{\rho \tau } \varphi \nabla {^{\alpha }}_{\sigma } \varphi + \nabla ^{\rho \alpha } \varphi \nabla {_{\sigma }}^{\tau } \varphi \right) \bigg ] \\
& \, + \partial _{\varphi } \hat{G}_5 \bigg [ \frac{1}{2}\delta ^{\alpha }_{\sigma } \nabla _{\rho } \varphi \nabla ^{\rho \tau } ... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.0232376791536808,
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-0.011412858963012695,
-0.03307897970080376,
0.0... |
0002bc6c8197ff94cd54efceb13499ea76fbeec5 | subsection | 40 | 63 | Conserved current coefficients for | \\
& \, \quad \left. + g^{\alpha \lambda } \left( \nabla {^{\kappa }}_{\kappa \rho } \varphi - \nabla {_{\rho \kappa }}^{\kappa } \varphi \right) \right) - \frac{1}{2}\nabla _{\rho } \Delta ^{\alpha }_{\sigma \lambda } \nabla ^{\rho \lambda } \varphi + \nabla _{\rho } \Delta ^{\lambda }_{\lambda \sigma } \nabla ^{\alph... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.004052017815411091,
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0.000835585524328053,
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-0.035865314304828644,
0... |
3a404ba52a95bbee898b52db30b63434095e80f2 | subsection | 41 | 63 | Conserved current coefficients for | \\
& \, \quad - \Delta ^{\rho }_{\rho \sigma } \nabla _{\lambda } \varphi \nabla ^{\alpha \lambda } \varphi + \Delta ^{\rho }_{\sigma \lambda } \Big ( \nabla _{\rho } \varphi \nabla ^{\alpha \lambda } \varphi - \nabla ^{\alpha } \varphi \nabla {_{\rho }}^{\lambda } \varphi + \nabla ^{\lambda } \varphi \nabla {_{\rho }}... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.007330211810767651,
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0.0006005164468660951,
-0.030679283663630486,
... |
b7d6c37bb34abb05aad77320c03c3c100c527e51 | subsection | 42 | 63 | Superpotential | The general formula for the superpotential reads [see Eq. (55) in Ref. petrovmain]\hat{i}^{\alpha \beta } = \left( \frac{2}{3} \bar{\nabla }_{\lambda } \hat{n}^{[\alpha \beta ]\lambda }_{\sigma } - \hat{m}^{[\alpha \beta ]}_{\sigma } \right) \xi ^{\sigma } - \frac{4}{3} \hat{n}^{[\alpha \beta ]\lambda }_{\sigma } \bar{... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
0.0019026355585083365,
0.022141147404909134,
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0.025376103818416595,
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0.0018539967713877559,
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0.043855033814907074,
0.027771804481744766,
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0.06396670639514923,
-0.03393653407692909,
0.0... |
3e92fb0e6dc3ef4265817220311f03cccf942ebe | subsection | 43 | 63 | Superpotential | Then, plugging () into (REF ), we get\hat{i}^{\alpha \beta }_{(3)} = \hat{i}^{\alpha \beta }_{(g)(3)} = 2 \hat{G}_3 \delta ^{[\alpha }_{\sigma } \nabla ^{\beta ]} \varphi \, \xi ^{\sigma }.For the Lagrangian \mathcal {L}_4 we split the superpotential into two parts – we will explicitly exclude the part originating from... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
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f9023553a1c7ff7d93efd4d4711f03d3cf185e9f | subsection | 44 | 63 | Superpotential | Generally speaking, if the structure of coefficients \hat{m}^{\alpha \tau }_{\sigma } and \hat{n}^{\alpha \tau \beta }_{\sigma } is\hat{m}^{\alpha \tau }_{\sigma } = F \, \hat{m}^{\alpha \tau }_{\sigma (EH)} + \hat{m}^{\alpha \tau }_{\sigma (rest)}, \qquad \hat{n}^{\alpha \tau \beta }_{\sigma } = F \, \hat{n}^{\alpha \... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
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bafe475ce507f89929bd6df2e052ae60e52c6d65 | subsection | 45 | 63 | Superpotential | The result turns out to be\hat{i}^{\alpha \beta }_{(5)(rest)} =& \, \Big [ \hat{G}_5 \Big ( 2 \delta ^{[\alpha }_{\sigma } \nabla {^{\beta ]\lambda }}_{\lambda } \varphi + 2 \nabla {^{[\alpha \beta ]}}_{\sigma } \varphi - 2 \delta ^{[\alpha }_{\sigma } \nabla {_{\lambda }}^{\beta ]\lambda } \varphi - \Delta ^{\rho }_{\... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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a47579cb3951fb5d658d61656b46a85c04b925d4 | subsection | 46 | 63 | Superpotential | \\
& \, \qquad + \delta ^{[\alpha }_{\sigma } \nabla ^{\beta ]} \varphi (\Box \varphi )^2 - \delta ^{[\alpha }_{\sigma } \nabla ^{\beta ]} \varphi \, \mbox{Tr} \, \Pi ^2 + 2 \nabla ^{[\alpha } \varphi \nabla {^{\beta ]}}_{\sigma } \varphi \, \Box \varphi \\
& \, \qquad \left. - 2 \nabla ^{[\alpha } \varphi \nabla ^{\be... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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7249dfd88f7963c577b417a7ecc9502525c34a69 | subsection | 47 | 63 | Superpotentials associated with nonlinear and linear perturbations | Superpotentials associated with the background are obtained simply by replacing all g_{\mu \nu } and \varphi by \bar{g}_{\mu \nu } and \bar{\varphi } and, consequently, all covariant derivatives \nabla are replaced by ones with respect to the background metric \bar{\nabla }, and the connections difference \Delta ^{\lam... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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ec434f1d8937f42218edf90eacac9742bba0d6cb | subsection | 48 | 63 | Superpotentials associated with nonlinear and linear perturbations | For the Lagrangian \mathcal {L}_4, we have the splitting (REF ), hence\bar{\hat{i}}^{\alpha \beta }_{(4)} = \bar{G}_4 \bar{\hat{i}}^{\alpha \beta }_{(EH)} + \bar{\hat{i}}^{\alpha \beta }_{(4)(rest)},where the second part of the expression is given by\bar{\hat{i}}^{\alpha \beta }_{(4)(rest)} = 4 \left[ \partial _X \bar{... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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8124b9192205950ec6d09f28c78baf3c39e72773 | subsection | 49 | 63 | Superpotentials associated with nonlinear and linear perturbations | For the last Lagrangian \mathcal {L}_5, we use the splitting (REF ), consequently, we have the expression for the background fields as follows:\bar{\hat{i}}^{\alpha \beta }_{(5)} = - \frac{1}{2}\bar{G}_5 \, \bar{\Box } \bar{\varphi } \, \bar{\hat{i}}^{\alpha \beta }_{(EH)} + \bar{\hat{i}}^{\alpha \beta }_{(5)(rest)},wi... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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2951e61a0004faea1352d34ccdc34e9dc3ab66cb | subsection | 50 | 63 | Superpotentials associated with nonlinear and linear perturbations | \\
& \, \qquad + \delta ^{[\alpha }_{\sigma } \bar{\nabla }^{\beta ]} \bar{\varphi } (\bar{\Box } \bar{\varphi })^2 - \delta ^{[\alpha }_{\sigma } \bar{\nabla }^{\beta ]} \bar{\varphi } \, \mbox{Tr} \, \bar{\Pi }^2 + 2 \bar{\nabla }^{[\alpha } \bar{\varphi } \bar{\nabla }{^{\beta ]}}_{\sigma } \bar{\varphi } \, \bar{\B... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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af47a7300de84bb98d794fbaba829308e215c3a2 | subsection | 51 | 63 | Superpotentials associated with nonlinear and linear perturbations | These will be subsequently used in the expressions for the linearized superpotential.The linearization of these superpotentials is done by assuming the metric and the scalar field in the form g_{\mu \nu } = \bar{g}_{\mu \nu } + \varepsilon h_{\mu \nu } and \varphi = \bar{\varphi } + \varepsilon \delta \varphi in superp... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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d74cf639d9ada6a8aaf803bb25dc643587471da6 | subsection | 52 | 63 | Superpotentials associated with nonlinear and linear perturbations | The linearization of the following expressions is obtained easily:G_i(\varphi , X) &= G_i(\bar{\varphi }, \bar{X}) + \varepsilon \left( \partial _{\varphi } G_i(\bar{\varphi }, \bar{X}) \delta \varphi + \partial _{X} G_i(\bar{\varphi }, \bar{X}) \delta X \right) + O(\varepsilon ^2), \\
\delta X &= - \bar{\nabla }^{\mu ... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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447b450b3e531dd617b412cd3119d3e0ff99e0c1 | subsection | 53 | 63 | Superpotentials associated with nonlinear and linear perturbations | \\
& \qquad \qquad \left. + \partial _X \bar{\hat{G}}_3 \bar{\nabla }^{\beta ]} \bar{\varphi } \left( \frac{1}{2}\delta g^{\mu \nu } \bar{\nabla }_{\mu } \bar{\varphi } \bar{\nabla }_{\nu } \bar{\varphi } - \bar{\nabla }_{\rho } \bar{\varphi } \bar{\nabla }^{\rho } \delta \varphi \right) \right] \xi ^{\sigma }.Assuming... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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3a740f2f4aa2f70e505ff27b45238556e2fc4b47 | subsection | 54 | 63 | Superpotentials associated with nonlinear and linear perturbations | In the case of the Lagrangian \mathcal {L}_4, the decomposition looks as follows:\delta \hat{i}^{\alpha \beta }_{(4)} = \delta G_4 \bar{\hat{i}}^{\alpha \beta }_{(EH)} + \bar{G}_4 \delta \hat{i}^{\alpha \beta }_{(EH)} + \delta \hat{i}^{\alpha \beta }_{(4)(rest)},and for the \delta \hat{i}^{\alpha \beta }_{(4)(rest)} we... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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995f88d6ecab1df021b7dd3dba39f37a98c97137 | subsection | 55 | 63 | Superpotentials associated with nonlinear and linear perturbations | It is the Einstein tensor G_{\mu \nu } and the third derivatives of scalar field \nabla _{\alpha \beta \gamma } \varphi . Let us examine them closer. | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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c60433accc931c86e257a4d7b4dd4eae194c96e7 | subsection | 56 | 63 | Superpotentials associated with nonlinear and linear perturbations | Using (REF ) and realizing that \Delta ^{\lambda }_{\mu \nu } is already of the first order, we obtain the Riemann tensor and, by contraction, the linearized Ricci tensorR{^{\lambda }}_{\tau \rho \sigma } &= \bar{R}{^{\lambda }}_{\tau \rho \sigma } + \varepsilon \left( \bar{\nabla }_{\rho } \delta \Delta ^{\lambda }_{\... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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3f7b826d9ec4d01b9642b55c7d17b59b8d1399eb | subsection | 57 | 63 | Superpotentials associated with nonlinear and linear perturbations | Converting the outermost derivative into a background one, we obtain\nabla _{\alpha \beta \gamma } \varphi = \bar{\nabla }_{\alpha }\left( \nabla _{\beta \gamma } \varphi \right) - \Delta ^{\rho }_{\alpha \beta } \nabla _{\gamma \rho } \varphi - \Delta ^{\rho }_{\gamma \alpha } \nabla _{\beta \rho } \varphi ,and after ... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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6df346ec76acb004eb75075963657af4fdcf1435 | subsection | 58 | 63 | Superpotentials associated with nonlinear and linear perturbations | The substitution 2 \bar{\nabla }_{[\alpha \beta ]\gamma } \delta \varphi = - \bar{R}{^{\rho }}_{\gamma \alpha \beta } \bar{\nabla }_{\rho } \delta \varphi completes the result, so that we have\delta \left( \nabla _{[\alpha \beta ]\gamma } \varphi \right) = - \frac{1}{2}\bar{R}{^{\rho }}_{\gamma \alpha \beta } \bar{\nab... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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f14b0cc566dc52e8a67b9c4b46e6cea7b8c29c13 | subsection | 59 | 63 | Brans-Dicke theory | Considering the Brans-Dicke LagrangianL = \sqrt{-g} \left( \frac{1}{2}\varphi R + \frac{\omega }{\varphi } X - U(\varphi ) \right),all of the results considerably simplify. The Brans-Dicke theory is a special case of a general Horndeski theory with Lagrangians \mathcal {L}_2 and \mathcal {L}_4 with functions K and G_4 ... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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54453fe12d399f5f759ff7926b043dca532bf80f | subsection | 60 | 63 | Conserved current coefficients for the Einstein-Hilbert Lagrangian | Considering the Einstein-Hilbert Lagrangian density, \hat{\mathcal {L}}_{(EH)} = \hat{R} = \sqrt{-g} \, R, we can calculate the coefficients using the formulas (REF )–() with \mathcal {L}_{(EH)}. Concerning \hat{u}^{\alpha }_{\sigma } we have to add the term \hat{\mathcal {L}}\delta ^{\alpha }_{\sigma } to () which we ... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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89e2a46339838f10b4eb092807932ec0e4f5be9b | subsection | 61 | 63 | Conserved current coefficients for the Einstein-Hilbert Lagrangian | For the derivative of the scalar curvature with respect to the second derivative of the metric, see (REF ), the derivative of the Ricci tensor w.r.t. the first derivative of the metric tensor can be found by contracting (REF ):\frac{\partial R}{\partial g_{\mu \nu |\alpha }} = \Delta ^{\alpha }_{\rho \kappa } \left( g^... | {
"cite_spans": []
} | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
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2b91e5caa6e0c960573847371b2e4be45f505b13 | subsection | 62 | 63 | Conserved current coefficients for the Einstein-Hilbert Lagrangian | (84). Considering the vector density \hat{d}^{\mu } = \hat{k}^{\mu } = \hat{g}^{\mu \rho } \Delta ^{\kappa }_{\rho \kappa } - \hat{g}^{\rho \kappa } \Delta ^{\mu }_{\rho \kappa } as in Ref. KBL, the following conserved current and superpotential modification, denoted as \hat{m}^{\alpha \beta }_{\sigma (k)} and \hat{i}^... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1103/physrevd.55.5957",
"end": 1049,
"openalex_id": "https://openalex.org/W2156125199",
"raw": "J. Katz, J. Bičák, and D. Lynden-Bell, “Relativistic conservation laws and integral constraints for large cosmological perturbations,” Phys. ... | 10.1063/1.5003190 | 1804.02298 | Covariant conserved currents for scalar-tensor Horndeski theory | [
"Josef Schmidt",
"Jiří Bičák"
] | [
"gr-qc",
"astro-ph.CO",
"hep-th"
] | 2,018 | en | Physics | [
-0.00631915545091033,
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0.040302254259586334,
-0.013469678349792957,
... |
4ae9f42c850471ec1c88971a6f91340e1befab0e | abstract | 0 | 74 | Abstract | Given a locally compact abelian group $G$, we give an explicit formula for
the Dixmier--Douady invariant of the $C^*$-algebra of the groupoid extension
associated to a \v{C}ech $2$-cocycle in the sheaf of germs of continuous
$G$-valued functions. We then exploit the blow-up construction for groupoids to
extend this to ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.029782777652144432,
-0.014670154079794884,
-0.01... | |
bf94baf4933636dced4b575772bfe5adbba917e4 | subsection | 1 | 74 | Introduction | This article provides explicit formulas for
the Dixmier–Douady invariants of a large class of continuous-trace
C^{*}-algebras arising from groupoid extensions. Continuous-trace
C^{*}-algebras are amongst the best understood and most intensively
studied classes of Type I C^{*}-algebras. A C^{*}-algebra A is a
continuous... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06152505427598953,
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0.02131708897650242,
-0.0010... | |
39bfa2e0b0094edc447b1d2479c110c33eb1b206 | subsection | 2 | 74 | Introduction | Consider a
second-countable locally compact Hausdorff space X and a
second-countable locally compact abelian group G. Take a Čech
2-cocycle c on X, relative to a locally finite open cover
{U}= \lbrace U_i : i \in I\rbrace of X, taking values in the sheaf {G} of
germs of continuous G-valued functions on X. The associate... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-0.02835702709853649,
... | |
ce057adcf6114816afe65fcb52e09adbc88ff2c9 | subsection | 3 | 74 | Introduction | A little more work puts us back in the situation
of Theorem REF , and we can use this to compute the
Dixmier–Douady invariant of C^{*}(\Sigma ^{\prime }). The blowup operation
determines an equivalence of extensions, and hence a Morita
equivalence of their C^{*}-algebras, yielding a computation of
\delta (C^{*}(\Sigma ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.009751016274094582,
-0.005... | |
a3abb98d86b7c326dced840509168d955b21ab7b | subsection | 4 | 74 | Central Isotropy | In the sequel, \Sigma will always be a second-countable locally
compact Hausdorff groupoid with a Haar system
\lbrace \lambda ^{u}\rbrace _{u\in \Sigma ^{(0)}}. The isotropy groupoid of
\Sigma is the closed subgroupoidI(\Sigma ) = \lbrace \,\gamma \in \Sigma :s(\gamma )=r(\gamma )\,\rbrace .Note that I(\Sigma ) is a gr... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-0.002962661674246192,
-0.02509010024368763,... | |
f52d91ad7d727ecdac3b5d62c4368c5839150529 | subsection | 5 | 74 | Central Isotropy | Recall that a
\mathbf {T}-groupoid \Sigma over a groupoid \mathcal {R} is a
unit-space-preserving groupoid extension\begin{}[column sep=3cm]
\Sigma ^{(0)}\times \mathbf {T}[r,"\iota ", hook] [dr,shift left, bend
right = 15] [dr,shift right, bend right = 15]&\Sigma [r,"\pi ", two heads] [d,shift left] [d,shift
right]&\m... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-0.02516907826066017,
... | |
1504c2043f5a69ddc94fc5a229215dad625a5b60 | subsection | 6 | 74 | Central Isotropy | However, the
isotropy groupoid I(\Sigma ) acts on the right and left of \Sigma ,
and with respect to the quotient topologies\mathcal {R}\cong I(\Sigma )\backslash \Sigma = \Sigma /I(\Sigma ).As observed above, I(\Sigma ) has a Haar system precisely when
u\mapsto \Sigma (u) is continuous. In this case, the orbit map
k:\... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.021807624027132988,
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-0.0064095184206962585,
-0.01785508729517... | |
b8b851d22182343396bc4beda2537538bddead1a | subsection | 7 | 74 | Central Isotropy | If
\lbrace \lambda ^{u}\rbrace _{u\in \Sigma ^{(0)}}, is a Haar system on \Sigma and
\mu is a Haar measure on \Sigma (u), then \operatorname{Ind}(u,\tau ) acts
by convolution on the completion of C_{c}(\Sigma _{u}) with respect
to to the pre-inner product
(f_{1}\mid f_{2})= \int _{\Sigma (u)} \int _{\Sigma }
\overlin... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.021776078268885612,
0.... | |
16e3b178cae7eac5c07b84095d9f2f4eb0563e49 | subsection | 8 | 74 | Central Isotropy | We record some elementary facts about
normalized cocycles for reference.
Lemma 3.1 Let c\in Z^{2}({U},{G}) be normalized. Then
for all i,j,k\in I,
c_{iij}(x)=c_{ijj}(x)=0.
c_{iji}(x)=c_{jij}(x).
c_{ijk}(x)=-c_{jik}(x) +c_{iji}(x).
c_{ijk}(x)=-c_{ikj}(x)+c_{jkj}(x).
c_{iji}(x)+c_{jki}(x)= -c_{ikj}(x)+c_{iki}(x)+c_{... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06261596828699112,
0.033444009721279144,
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-0.0028931661508977413,
... | |
58b6e0634ac7a2ae3a27055102acc3bd7a3001a3 | subsection | 9 | 74 | Central Isotropy | Since
\bigl (g,(i,x,j)\bigr )\bigl (g,(i,x,j)\bigr )^{-1}
=
\bigl (g,(i,x,j)\bigr ) \bigl (-g-c_{iji}(x),(j,x,i)\bigr )
= \bigl (0
,(i,x,i)\bigr ),
and similarly
\bigl (g,(i,x,j)\bigr )^{-1}\bigl (g,(i,x,j)\bigr )=\bigl (0,(j,x,j)\bigr ),
we can identify the unit space of \Sigma _{c} with \coprod U_{i}. Let \mu be ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06767363101243973,
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-0.025629326701164246,
... | |
b5c22de7a398b001c6d5b4e2944786b9000c9ead | subsection | 10 | 74 | Central Isotropy | As in Remark REF , we can identify the
spectrum of C^{*}(\Sigma _{c}) as a set with \widehat{G}\times X via
(\tau ,x) \mapsto [\operatorname{Ind}((i,x),\tau )] for any i such that
x\in U_{i}.
Lemma 3.2 Let \Sigma _{c} be as above. Let
I(x)=\lbrace \,j\in I:x\in U_{j}\,\rbrace . Then \operatorname{Ind}((i,x),\tau ) is
e... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04016399011015892,
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0.0... | |
84391d6c617f3d7e584ebffc72dfcb8a618e4409 | subsection | 11 | 74 | Central Isotropy | But
U(&f_{1}*f_{2})(j) \\
&=
\int _{G} f_{1}*f_{2}(g-c_{iji}(x), (j,x,i))\tau (g)\,d\mu (g) \\
&= \sum _{k} \int _{G}\int _{G}
f_{1}(h,(j,x,k)) f_{2}(-h+g-c_{iji}(x) -c_{jki}(x),(k,x,i))
\tau (g)\,d\mu (h)\,d\mu (g) \\
{which, since c_{iji}(x)+c_{jki}(x) =
c_{ijk}(x)+c_{iki}(x), is}
&=\sum _{k} \int _{G}\int _{G}
f_{1... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-0.002789803547784686,
-0.0053622196428477... | |
218512e3dcc10eaf9321f4c930ec4fbc803cc446 | subsection | 12 | 74 | Central Isotropy | Theorem 3.4 Suppose that X is a second-countable
locally compact Hausdorff space and that G is a second-countable
locally compact abelian group. Let {G} be the sheaf of germs of
continuous G-valued functions on X. Suppose that c\in Z^{2}({U},{G}) is a
normalized cocycle on a locally finite cover {U} by precompact
open ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-0.020725451409816742,
0.... | |
c66c7e6a73a34f1612faf6f40a244d2d89f0ffdd | subsection | 13 | 74 | Central Isotropy | To construct A(\nu ), we begin by forming the algebra A_{1}(\nu )
which is the set of sparse I\times I-matrices
f=\bigl (f_{ij}\bigr )_{i,j\in I}
where each f_{ij}\in C_{0}(X) and vanishes off
U_{ij}.Requiring that each f_{ij} belongs to C_{0}(X)
rather than just to C(X) is redundant if each U_{ij} is
precompact as w... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03401653841137886,
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-0.01076936163008213,
0... | |
9b80c9498a90d63f31e60ab2fee5f15baaa0de15 | subsection | 14 | 74 | Central Isotropy | To see
that it is anti-multiplicative, we require
Lemma REF (e):
(f*g)^{*}_{ij}(x)
&= \nu _{iji}(x) \overline{(f*g)_{ji}(x)}\\
&= \nu _{iji}(x) \sum _{k}\nu _{jki}(x) \overline{f_{jk}(x)}
\overline{g_{ki}(x)} \\
{which, using Lemma~\ref {lem-norm-coc}(e), is}
&= \sum _{k}\overline{\nu _{ikj}(x)} \nu _{iki}(x) \overli... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-... | |
ff99c261abd11aa88104b3e9bbf2f94be1960da2 | subsection | 15 | 74 | Central Isotropy | The groupoid is the
blow-up \Gamma _{{U}} associated to the cover {U} of X corresponding to
\nu
and the cocycle \varphi _{\nu } in Z^{2}(\Gamma _{{U}},\mathbf {T}) is given by
\varphi _{\nu }\bigl ((i,x,j),(j,x,k)\bigr )=\overline{\nu _{ijk}(x)}.
(The complex conjugate in (REF ) is missing from the formula
in .) Not... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06275147944688797,
0.02754533477127552,
-0.05628099665045738,
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-0.008515399880707264,
... | |
63dccf79a9203aab5d6e8d2d95c0d711102907d2 | subsection | 16 | 74 | Central Isotropy | So in this section we show that
C^{*}(\Gamma _{,\varphi _{\nu ^{c}}) is continuous-trace with the desired Dixmier--Douady
class for the given cover , and then use this to
prove Theorem~\ref {thm-main-dd-calc}.
}We let A_{1}(\nu ^{c}) be defined just as in
Section~\ref {sec:raeb-tayl-algebra}. We want to define
(f_{ij})... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.014136127196252346,
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0.05346248298883438,
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0.0033375907223671675,
0.... | |
47973d21dee28bc1bbb8c3a94f6310e6b97e2942 | subsection | 17 | 74 | Central Isotropy | For this, it
suffices to consider each summand
\begin{equation}
a_{j}(\tau ,x)=
{\left\lbrace \begin{array}{ll}
f_{ij}(\tau ,x)g_{jk}(\tau ,x)
\overline{\nu ^{c}_{ijk}(\tau ,x)}&\text{if
$x\in U_{ijk}$}\\
0&\text{otherwise.}
\end{array}\right.}
\end{equation}
We clearly have h_{ik}(\tau _n, x_n) \rightarrow h_{ik}(\tau... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.04612571746110916,
0.045118339359760284,
-0.0137064503505826,
-0.01726280152797699,
-0.04047829285264015,
0.01... | |
e6b92dcff58742e78e6a10e84c5411c5087bdc4c | subsection | 18 | 74 | Central Isotropy | \begin{}
The C^{*}-algebra A(\nu ^{c}) has continuous
trace with spectrum \widehat{G}\times X and Dixmier-Douady class
\delta (A(\nu ^{c}))=[\nu ^{c}].
\end{}
\begin{}
We have already seen that A(\nu ^{c}) is a C^{*}-algebra with Hausdorff
spectrum. We continue by making the necessary modifications to the
proof of \ci... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.07255551218986511,
0.023005405440926552,
-0.02942800149321556,
-0.021006925031542778,
0.0065446412190794945,
-0.04558365419507027,
-0.011800186708569527,
0.00919148325920105,
0.05690329149365425,
0.04573620855808258,
0.020076334476470947,
-0.010816201567649841,
-0.0013186536962166429,
0... | |
5350f88a6798b6081cea4dde437ba33d8693e88b | subsection | 19 | 74 | Central Isotropy | Similarly,
let \phi _{(n,i)(m,j)} \in C_{c}^{+}(W_{(n,i)(m,j)}) be identically
one on F_{(n,i)(m,j)}. Then we get v((n,i),(m,j))\in A(\nu ^{c})
with
v((n,i),(m,j))_{rs}(\tau ,x)=
{\left\lbrace \begin{array}{ll}
\phi _{(n,i)(m,j)}(\tau ,x)&\text{if $r=i$ and $s=j$, and} \\ 0
&\text{otherwise.}
\end{array}\right.}
The... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.043251171708106995,
0.022602323442697525,
-0.022785460576415062,
-0.009675809182226658,
0.00563913444057107,
0.006974518299102783,
-0.002293044701218605,
0.041908156126737595,
0.03076723963022232,
0.0378180667757988,
-0.00725304102525115,
-0.029088472947478294,
-0.01634509675204754,
0.0... | |
09e2597d855843834b00eae51c56a1563ecb61c9 | subsection | 20 | 74 | Central Isotropy | Thus
\bigl [v((n,i),
&(m,j))v((m,j),(l,k))\bigr ]_{rs}(\tau ,x) \\
&= \sum _{a} \overline{\nu ^{c}_{rsa}(\tau ,x)}
v((n,i),(m,j))_{ra}(\tau ,x) v ((m,j),(l,k))_{as}(\tau ,x) \\
&=
{\left\lbrace \begin{array}{ll}
0&\text{if $r\ne i$ or $s\ne l$, and } \\
\overline{\nu ^{c}_{ijk}(\tau ,x)}\phi _{(n,i)(m,j)}(\tau ,x)
\p... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.027297206223011017,
0.027266688644886017,
-0.031432222574949265,
-0.02239927276968956,
-0.008353952318429947,
0.027220914140343666,
-0.01512866374105215,
0.015471977181732655,
0.02490164339542389,
0.02078188583254814,
-0.013038268312811852,
-0.0086896363645792,
0.001400908106006682,
0.0... | |
7d5a4f2e87e2c032e151a84197a17e7bcbb6f13f | subsection | 21 | 74 | Central Isotropy | Since
finite sums of such functions are dense in the inductive limit
topology, we deduce that \Phi (f)\in A(\nu ^{c}) for all f.
Note
\Phi (f^{*}(i,(\tau ,x),j))
&= \int _{G}\tau (g)
\overline{f(-g-c_{iji}(x),(j,x,i))}
\,d\mu (g) \\
&= \overline{\int _{G} \tau (g) f(g-c_{iji}(x),(j,x,i))\,d\mu (g)} \\
&= \overline{\t... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04732941836118698,
0.00990225374698639,
0.00975730549544096,
-0.02636532299220562,
-0.015372143127024174,
-0.05447002500295639,
0.013785341754555702,
0.01846182905137539,
0.00805607158690691,
0.037503451108932495,
-0.05392074957489967,
0.014128640294075012,
0.01438802108168602,
0.020842... | |
de79f035de52ba666c98928493ed6cd13f26e3ea | subsection | 22 | 74 | Central Isotropy | It follows from Lemma REF , that
\operatorname{Ind}((i,x),\tau )(f) is equivalent to multiplication by the
matrix
\bigl [ \tau (c_{ijk})(x)\Phi (f) \bigr ].
Since \tau (c_{ijk}(x))=\overline{\nu ^{c}_{ijk}(\tau ,x)}, we see that
\operatorname{Ind}((i,x),\tau )(f)=\pi _{(i,(\tau ,x))}(\Phi (f)).
This shows immediat... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03514743968844414,
0.02521645650267601,
-0.018138153478503227,
-0.012524326331913471,
0.02454523742198944,
-0.04686325043439865,
0.011670048348605633,
-0.006818969268351793,
0.04030361771583557,
0.03624579682946205,
-0.018763607367873192,
0.010678475722670555,
-0.03197440505027771,
0.02... | |
bfed04ed6c5b45287b8722ce607f62653639e5e5 | subsection | 23 | 74 | Central Isotropy | If \pi has a continuous section \kappa :R(\psi )\rightarrow \Sigma (this is
equivalent to \pi being trivial as a principal G-bundle), then
Proposition REF shows that \Sigma is
properly isomorphic to the extension \Sigma ({R(\psi )}, {\varphi })
constructed from a continuous (normalised) G-valued 2-cocycle
\varphi \in Z... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.07947389036417007,
0.04590196907520294,
0.008248009718954563,
0.022218506783246994,
0.0220811665058136,
-0.04782472550868988,
0.0345790758728981,
-0.0004690054338425398,
0.034487515687942505,
0.04904552176594734,
-0.027651052922010422,
0.004036260303109884,
0.027010135352611542,
0.04507... | |
c73b761be86d2d19e20ae71de0755283afa2d64a | subsection | 24 | 74 | Central Isotropy | Lemma 6.1 If y\in U and \tau \in \widehat{G}, then
\mathop {\Sigma _{U}\mathord {\mathop {\text{--}}}}\!\operatorname{Ind}\nolimits (\operatorname{Ind}^{\Sigma (U)}(y,\tau )) is equivalent to
\operatorname{Ind}^{\Sigma } (y,\tau ).
As in *p. 12 or
*Equation 1.3, the
C_{c}(\Sigma ({U}))-valued inner product on C_{c}(\S... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.00982897449284792,
-0.0037297448143363,
-0.027197567746043205,
-0.023839842528104782,
0.02922746352851391,
-0.054059360176324844,
-0.0329056978225708,
0.04389461502432823,
0.01901692897081375,
0.0349813811480999,
-0.015285276807844639,
-0.01347668468952179,
-0.014773987233638763,
-0.016... | |
eac428c4c619425c9814fefc08ac676433cac6db | subsection | 25 | 74 | Central Isotropy | Then \mathop {\Sigma _{U}\mathord {\mathop {\text{--}}}}\!\operatorname{Ind}\nolimits \operatorname{Ind}^{\Sigma (U)}(y,\tau ) acts by
convolution on the completion of
C_{c}(\Sigma _{U})\odot C_{c}(\Sigma (U)_{y}) with respect to the
inner product
\bigl (f_{1}\otimes k_{1}
& \mid f_{2}\otimes k_{2}\bigr )
= \bigl (\l... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.007268034853041172,
0.012246925383806229,
-0.004700377583503723,
-0.005047564394772053,
0.006150169298052788,
-0.04163191467523575,
-0.03714518994092941,
0.006569845601916313,
0.012079054489731789,
0.034581348299980164,
-0.011926445178687572,
0.02141113579273224,
-0.007012413814663887,
... | |
5b9622f9c16ff8006170c0cd2ead67f7df071608 | subsection | 26 | 74 | Central Isotropy | It follows that W defines an isometry from the space of
\mathop {\Sigma _{U}\mathord {\mathop {\text{--}}}}\!\operatorname{Ind}\nolimits (\operatorname{Ind}^{\Sigma (U)}(y,\tau ) into the space of
\operatorname{Ind}^{\Sigma }(y,\tau ) that intertwines the two representations.
Since the representations are irreducible, ... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.005053368862718344,
0.026102080941200256,
-0.003909210208803415,
-0.020457563921809196,
-0.004321107175201178,
-0.026727555319666862,
0.017787858843803406,
0.014538447372615337,
0.037162285298109055,
0.04143381118774414,
-0.001726726652123034,
-0.006289060693234205,
-0.005396616645157337,... | |
58bac4c12d00de6996a36227d5d6b66ccf82c100 | subsection | 27 | 74 | Central Isotropy | The C_{c}(\Sigma ^{\prime })-valued inner product on C_{c}(Z) is given by
\langle f_{1}\mathrel {,}f_{2}
\rangle _{{\hspace{-1.66656pt}}\copy
\scriptscriptstyle {\Sigma ^{\prime }}}(i,\gamma ,j)=\int _{\Sigma }
\overline{f_{1}(i,\sigma ^{-1})}
f_{2}(j,\sigma ^{-1}\gamma )\,d\lambda ^{r(\gamma )} (\sigma ). | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03134506940841675,
0.017412232235074043,
-0.015840400010347366,
0.006397964898496866,
0.022310853004455566,
-0.03155871480703354,
0.008706116117537022,
0.027270514518022537,
-0.020006516948342323,
0.02356221340596676,
-0.03155871480703354,
0.007351746316999197,
-0.015275761485099792,
-0... | |
1d215e764ae42236c24d78e8e82c5fe53dcdfa7d | subsection | 28 | 74 | Central Isotropy | Thus \mathop {Z\mathord {\mathop {\text{--}}}}\!\operatorname{Ind}\nolimits
(\operatorname{Ind}^{\Sigma ^{\prime }}((i,y),\tau ) acts by convolution on the completion of
C_{c}(Z)\odot C_{c}(\Sigma ^{\prime }_{(i,y)}) with respect to the inner product
\bigl (f_{1}\otimes k_{1}
& \mid f_{2}\otimes k_{2}\bigr )
= \bigl... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.013592226430773735,
0.0005657068104483187,
0.006498473230749369,
-0.010386873036623001,
0.013714335858821869,
-0.02672654576599598,
-0.02048373781144619,
0.01393565721809864,
0.00808970257639885,
0.054552070796489716,
-0.022223787382245064,
0.0018163672648370266,
-0.026451801881194115,
... | |
1728f71bc14b497dd0f24802a8c4fb2ea1ac9d1d | subsection | 29 | 74 | Central Isotropy | As in the proof of Lemma REF , W extends to an
intertwining
unitary implementing the desired equivalence.
Theorem 6.3 Let Y and X be second-countable
locally compact Hausdorff spaces with X locally G-trivial as
defined above. Suppose that \psi :Y\rightarrow X is a local homeomorphism,
and let \Sigma be a groupoid exten... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.048133980482816696,
0.04166321083903313,
-0.00006062769898562692,
-0.02299870364367962,
0.007512350101023912,
-0.03096507489681244,
0.01479578111320734,
-0.01165196392685175,
0.041235897690057755,
0.027424465864896774,
-0.012437918223440647,
0.023059748113155365,
-0.0030522500164806843,
... | |
7fb642674ab1450789e0bbcdd2a861f029aeb0a7 | subsection | 30 | 74 | Central Isotropy | If we let
R^{\prime } = \lbrace \,(i,(x,y),j):\text{$\psi (x)=\psi (y)$, $x\in V_{i}$ and
$y\in V_{j}$}\,\rbrace ,
then we obtain a generalised twist
\begin{}[column sep=3cm]
G \times \coprod V_{j} [r,"\iota ^{\prime }", hook]
[dr,shift left, bend right = 15] [dr,shift right,
bend right = 15]&\Sigma ^{\prime } [r,"... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03258601576089859,
0.016155706718564034,
-0.009191513992846012,
0.023081760853528976,
0.0071815853007137775,
-0.010602659545838833,
-0.018825441598892212,
0.011212884448468685,
0.018535584211349487,
0.025476893410086632,
-0.04637710005044937,
-0.0024180165491998196,
-0.026468507945537567,... | |
6f1136a23f79dc9e44987c2dabdc26c505598a3d | subsection | 31 | 74 | Central Isotropy | There is a
groupoid homomorphism \tau :R^{\prime }\rightarrow \Gamma _{W} such that
\tau (i,(x,y),j)=(i,\psi (x),j)
and
\tau ^{-1}(i,w,j)=(i,(x,y),j)\quad \text{if $x\in V_{i}$ and $y\in V_{j}$ satisfy $\psi (x) = w = \psi (y)$.}
So, defining
\tilde{\varphi } := \varphi \circ (\tau ^{-1} \times \tau ^{-1}) \in Z^{2... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05635873228311539,
0.002076836070045829,
-0.017835235223174095,
-0.006194273009896278,
0.0037798797711730003,
-0.04451943188905716,
0.04137652367353439,
0.014219366014003754,
0.020474666729569435,
0.048181068152189255,
0.009367694146931171,
-0.0038771419785916805,
-0.018247168511152267,
... | |
0f863f597839ec655ac5ffc2d519a913c29e0577 | subsection | 32 | 74 | Central Isotropy | We summarize all of this by drawing
the diagram
\begin{}[column sep=3cm]
\Gamma ^{(0)} \times G [r,"\iota ", hook] [dr,shift
left, bend right = 15] [dr,shift right, bend right =
15]&\Sigma [r,"\pi ", two heads] [d,shift left]
[d,shift right]&\Gamma [dl,shift left, bend left = 15]
[dl,shift right, bend left = 15]
\\
&\... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.029356233775615692,
0.02462627924978733,
-0.024412669241428375,
0.005862854886800051,
0.009612488560378551,
-0.036252811551094055,
0.01953013427555561,
0.0036237554159015417,
0.0024412667844444513,
0.03265194594860077,
-0.017943311482667923,
-0.016814226284623146,
-0.027296414598822594,
... | |
93837d999e46ddf87f9ffea76e605527108d0ab7 | subsection | 33 | 74 | Central Isotropy | As in , the Baer sum \Sigma _{1}\ast \Sigma _{2} of two
extensions
\Gamma ^{(0)}\times G\longrightarrow \Sigma _{i}\longrightarrow \Gamma
is the extension
\Gamma ^{(0)}\times G\longrightarrow \Sigma _{1}\ast \Sigma _{2}\longrightarrow \Gamma with
\Sigma _{1}\ast \Sigma _{2}=\lbrace (\sigma _{1},\sigma _{2})\in \Sigma... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.045808158814907074,
0.012161592952907085,
-0.04409912973642349,
0.035340338945388794,
0.017258170992136,
-0.05935834348201752,
0.03906358778476715,
0.058015529066324234,
0.030716797336935997,
0.028092212975025177,
-0.029755467548966408,
0.0024605481885373592,
-0.01442758645862341,
-0.00... | |
4208fdcce2cb24f5ca1544c5512d469f21da23c6 | subsection | 34 | 74 | Central Isotropy | A continuous 2-cocycle \varphi :\Gamma ^{(2)}\rightarrow G is a continuous
function such that for all
(\gamma _{0},\gamma _{1},\gamma _{2})\in \Gamma ^{(3)},
\varphi (\gamma _{0},\gamma _{1}) +
\varphi (\gamma _{0}\gamma _{1},\gamma _{2}) =
\varphi (\gamma _{1},\gamma _{2}) +
\varphi (\gamma _{0},\gamma _{1}\gamma _{2... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.034245118498802185,
-0.024569803848862648,
-0.02246382087469101,
-0.012933483347296715,
0.016924168914556503,
-0.04437825456261635,
0.023928852751851082,
0.05716675892472267,
-0.02266220934689045,
0.02037309855222702,
-0.015962742269039154,
0.021838130429387093,
0.007000867743045092,
0.... | |
861784c8e17368286624cda417b7cd028ee7260a | subsection | 35 | 74 | Central Isotropy | We have
(g,\gamma )^{-1} = (-g - \varphi (\gamma ^{-1}, \gamma ),
\gamma ^{-1}). The maps
\iota _{\varphi } : \Gamma ^{(0)} \times G \rightarrow \Sigma ({\Gamma }, {\varphi }) and
\pi _{\varphi } : \Sigma ({\Gamma }, {\varphi }) \rightarrow \Gamma are defined
by \iota _{\varphi }(u,g) = (g,u) and
\pi _{\varphi }(g, \ga... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.047545433044433594,
-0.016555454581975937,
-0.026122523471713066,
-0.0037326065357774496,
0.019027328118681908,
-0.04485994204878807,
0.0365288108587265,
0.04113687202334404,
0.014678660780191422,
0.033202461898326874,
-0.00010227951861452311,
-0.007018902339041233,
0.0008706871885806322,... | |
10bcaf4c4a76ba1bf3a7d8f184be5cca25b4dacb | subsection | 36 | 74 | Central Isotropy | Define \varphi :\Gamma ^{(2)}\rightarrow G by
\iota (r(\gamma _{1}),\varphi (\gamma _{1},\gamma _{2})) =
\tau (\gamma _{1})\tau (\gamma _{2})\tau (\gamma _{1}\gamma _{2})^{-1}. Then
\varphi is continuous. To see that \varphi is a 2-cocycle, first
note that
\iota (r(\gamma _{1}), \varphi (\gamma _{1}, \gamma _{2}))
\tau... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.02769014611840248,
0.0006436242838390172,
-0.026317082345485687,
0.014241736382246017,
0.017636258155107498,
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0.040276575833559036,
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0.009962350130081177,
-0.014996922574937344,
-0.... | |
d0c872f75053850710bfe38a6bb18c3576be4034 | subsection | 37 | 74 | Central Isotropy | Amer. Math. Soc.,
volume=343,
pages=117133,
ech:mams96article
author=Echterhoff, Siegfried,
title=Crossed products with continuous trace,
date=1996,
ISSN=0065-9266,
journal=Mem. Amer. Math. Soc.,
volume=123,
number=586,
pages=iviii, 1134,
review=98f:46055,
gre:pjm77article
author=Green, Philip,
title=C^*-algebras of tr... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03247417137026787,
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0.062262509018182755,
-0.006821407470852137,
0.0071189855225384235,
0.01974698156118393,
... | |
3c5ad56e905c89b28f516bc1e9aae428aa434b42 | subsection | 38 | 74 | Central Isotropy | Soc.,
volume=348,
number=9,
pages=36213641,
review=MR1348867 (96m:46125),
muhwil:jams04article
author=Muhly, Paul S.,
author=Williams, Dana P.,
title=The Dixmier-Douady class of groupoid crossed products,
date=2004,
ISSN=1446-7887,
journal=J. Aust. Math. Soc.,
volume=76,
number=2,
pages=223234,
review=MR2041246 (2005e:... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05035776644945145,
-0.010338601656258106,
-0.049747366458177567,
0.010819289833307266,
0.0001889608392957598,
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0.0008454954368062317,
0.06268778443336487,
-0.02038726396858692,
-0.0009141651098616421,
0.028032489120960236,... | |
90273d9ed93a4d59848a0f4b63d0dcc844668ec2 | subsection | 39 | 74 | Central Isotropy | Math.,
volume=45,
number=5,
pages=10321066,
review=94k:46141,
rw:moritabook
author=Raeburn, Iain,
author=Williams, Dana P.,
title=Morita equivalence and continuous-trace C^*-algebras,
series=Mathematical Surveys and Monographs,
publisher=American Mathematical Society,
address=Providence, RI,
date=1998,
volume=60,
ISBN=... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.030796976760029793,
-0.008607281371951103,
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-0.008393625728785992,
0.004860672168433666,
-... | |
ee346a8771c91b7c620bb042397d38e0ff0325f5 | subsection | 40 | 74 | A Class of Examples | Let X be a second-countable locally compact Hausdorff space and
G a second-countable locally compact abelian group. Let {G} be
the sheaf of germs of G-valued functions on X (see
*§4.1). Let \mathfrak {a} be an element in the sheaf
cohomology group H^{2}(X,{G}). Then \mathfrak {a} is represented by a
two cocycle c\in Z^... | {
"cite_spans": []
} | 1801.00832 | The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with
Continuous Trace | [
"Marius Ionescu",
"Alex Kumjian",
"Aidan Sims",
"Dana P. Williams"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.07708150148391724,
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0.009665703400969505,
0.0015934677794575691,
0.0034787377808243036,
0.0008472747867926955,
0.0... |
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