SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle
Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-12-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/10/1/2 |
| _version_ | 1797342458077511680 |
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| author | Viktor Abramov Olga Liivapuu |
| author_facet | Viktor Abramov Olga Liivapuu |
| author_sort | Viktor Abramov |
| collection | DOAJ |
| description | Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>3</mn></msub></semantics></math></inline-formula>-skew-symmetric covariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity <i>q</i>, and in each subspace, there is an irreducible representation of the rotation group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo><mo>↪</mo><mrow><mi>SU</mi></mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We find two <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-invariants of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>3</mn></msub></semantics></math></inline-formula>-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We study the invariant properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> and find its stabilizer. Making use of these invariant properties, we define an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-irreducible geometric structure and find its curvature and torsion. |
| first_indexed | 2024-03-08T10:33:34Z |
| format | Article |
| id | doaj.art-917681edca1c4b0a8f41ec33db276ba8 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-08T10:33:34Z |
| publishDate | 2023-12-01 |
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| series | Universe |
| spelling | doaj.art-917681edca1c4b0a8f41ec33db276ba82024-01-26T18:42:53ZengMDPI AGUniverse2218-19972023-12-01101210.3390/universe10010002SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion PrincipleViktor Abramov0Olga Liivapuu1Institute of Mathematics and Statistics, University of Tartu, 51009 Tartu, EstoniaInstitute of Forestry and Engineering, Estonian University of Life Sciences, 51006 Tartu, EstoniaMotivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>3</mn></msub></semantics></math></inline-formula>-skew-symmetric covariant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity <i>q</i>, and in each subspace, there is an irreducible representation of the rotation group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo><mo>↪</mo><mrow><mi>SU</mi></mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We find two <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-invariants of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>3</mn></msub></semantics></math></inline-formula>-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We study the invariant properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>z</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> and find its stabilizer. Making use of these invariant properties, we define an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>SO</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>-irreducible geometric structure and find its curvature and torsion.https://www.mdpi.com/2218-1997/10/1/2tensorsrotation grouptensor irreducible representationscomplex manifoldsspecial geometriesconnection |
| spellingShingle | Viktor Abramov Olga Liivapuu SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle Universe tensors rotation group tensor irreducible representations complex manifolds special geometries connection |
| title | SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle |
| title_full | SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle |
| title_fullStr | SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle |
| title_full_unstemmed | SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle |
| title_short | SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle |
| title_sort | so 3 irreducible geometry in complex dimension five and ternary generalization of pauli exclusion principle |
| topic | tensors rotation group tensor irreducible representations complex manifolds special geometries connection |
| url | https://www.mdpi.com/2218-1997/10/1/2 |
| work_keys_str_mv | AT viktorabramov so3irreduciblegeometryincomplexdimensionfiveandternarygeneralizationofpauliexclusionprinciple AT olgaliivapuu so3irreduciblegeometryincomplexdimensionfiveandternarygeneralizationofpauliexclusionprinciple |