Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras
We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>=</...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/8/1309 |
| _version_ | 1797522040932007936 |
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| author | Jerzy Lukierski |
| author_facet | Jerzy Lukierski |
| author_sort | Jerzy Lukierski |
| collection | DOAJ |
| description | We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> quantum inhomogeneous conformal Hopf algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">U</mi><mi>θ</mi></msub><mrow><mo>(</mo><mi>s</mi><mi>u</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>⋉</mo></mrow><msup><mi>T</mi><mn>4</mn></msup></mrow></semantics></math></inline-formula>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">U</mi><mover accent="true"><mi>θ</mi><mo stretchy="false">¯</mo></mover></msub><mrow><mo>(</mo><mi>s</mi><mi>u</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>⋉</mo></mrow><msup><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover><mn>4</mn></msup></mrow></semantics></math></inline-formula>), where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>T</mi><mn>4</mn></msup></semantics></math></inline-formula> describes complex twistor coordinates and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover><mn>4</mn></msup></semantics></math></inline-formula> the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> Heisenberg-conformal algebra (HCA) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>u</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>⋉</mo><msubsup><mi>H</mi><mo>ℏ</mo><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></msubsup></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mo>ℏ</mo><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></msubsup><mo>=</mo><msup><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover><mn>4</mn></msup><msub><mo>⋉</mo><mo>ℏ</mo></msub><msub><mi>T</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula> is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>λ</mi><mi>p</mi></msub></semantics></math></inline-formula>, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>H</mi><mo>ℏ</mo><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></msubsup></semantics></math></inline-formula>. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">U</mi><mi>θ</mi></msub><mrow><mo>(</mo><mi>s</mi><mi>u</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>⋉</mo><msup><mi>T</mi><mn>4</mn></msup><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> |
| first_indexed | 2024-03-10T08:21:00Z |
| format | Article |
| id | doaj.art-e7839a8d80564ea3b8f5a9f27eb26a96 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T08:21:00Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-e7839a8d80564ea3b8f5a9f27eb26a962023-11-22T09:59:10ZengMDPI AGSymmetry2073-89942021-07-01138130910.3390/sym13081309Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR AlgebrasJerzy Lukierski0Institute for Theoretical Physics, University of Wroclaw, Pl. Maxa Borna 9, 50-205 Wroclaw, PolandWe construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> quantum inhomogeneous conformal Hopf algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">U</mi><mi>θ</mi></msub><mrow><mo>(</mo><mi>s</mi><mi>u</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>⋉</mo></mrow><msup><mi>T</mi><mn>4</mn></msup></mrow></semantics></math></inline-formula>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">U</mi><mover accent="true"><mi>θ</mi><mo stretchy="false">¯</mo></mover></msub><mrow><mo>(</mo><mi>s</mi><mi>u</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>⋉</mo></mrow><msup><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover><mn>4</mn></msup></mrow></semantics></math></inline-formula>), where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>T</mi><mn>4</mn></msup></semantics></math></inline-formula> describes complex twistor coordinates and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover><mn>4</mn></msup></semantics></math></inline-formula> the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> Heisenberg-conformal algebra (HCA) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>u</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>⋉</mo><msubsup><mi>H</mi><mo>ℏ</mo><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></msubsup></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mo>ℏ</mo><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></msubsup><mo>=</mo><msup><mover accent="true"><mi>T</mi><mo stretchy="false">¯</mo></mover><mn>4</mn></msup><msub><mo>⋉</mo><mo>ℏ</mo></msub><msub><mi>T</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula> is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>λ</mi><mi>p</mi></msub></semantics></math></inline-formula>, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>H</mi><mo>ℏ</mo><mrow><mn>4</mn><mo>,</mo><mn>4</mn></mrow></msubsup></semantics></math></inline-formula>. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">U</mi><mi>θ</mi></msub><mrow><mo>(</mo><mi>s</mi><mi>u</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow><mo>⋉</mo><msup><mi>T</mi><mn>4</mn></msup><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula>https://www.mdpi.com/2073-8994/13/8/1309quantum deformationsquantum gravityclassical and quantum twistor geometry |
| spellingShingle | Jerzy Lukierski Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras Symmetry quantum deformations quantum gravity classical and quantum twistor geometry |
| title | Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras |
| title_full | Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras |
| title_fullStr | Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras |
| title_full_unstemmed | Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras |
| title_short | Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras |
| title_sort | palatial twistors from quantum inhomogeneous conformal symmetries and twistorial dsr algebras |
| topic | quantum deformations quantum gravity classical and quantum twistor geometry |
| url | https://www.mdpi.com/2073-8994/13/8/1309 |
| work_keys_str_mv | AT jerzylukierski palatialtwistorsfromquantuminhomogeneousconformalsymmetriesandtwistorialdsralgebras |