The Real Forms of the Fractional Supergroup SL(2,C)
The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfence...
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MDPI AG
2021-04-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/9/933 |
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| author | Yasemen Ucan Resat Kosker |
| author_facet | Yasemen Ucan Resat Kosker |
| author_sort | Yasemen Ucan |
| collection | DOAJ |
| description | The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> is one of these important groups. There are real forms of the classical Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> and the quantum group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mi>N</mi></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula>, for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mn>1</mn></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mn>2</mn></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula>. |
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| spelling | doaj.art-e1fdebb4229c445ba49fc882785ce2e42023-11-21T16:41:32ZengMDPI AGMathematics2227-73902021-04-019993310.3390/math9090933The Real Forms of the Fractional Supergroup SL(2,C)Yasemen Ucan0Resat Kosker1Department of Mathematical Engineering, Faculty of Chemistry and Metallurgy, Davutpasa Campus, Yildiz Technical University, Esenler, 34220 Istanbul, TurkeyDepartment of Mathematical Engineering, Faculty of Chemistry and Metallurgy, Davutpasa Campus, Yildiz Technical University, Esenler, 34220 Istanbul, TurkeyThe real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> is one of these important groups. There are real forms of the classical Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> and the quantum group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mi>N</mi></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula>, for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mn>1</mn></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mn>2</mn></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/9/9/933fractional supergroupHopf algebrastar-algebra |
| spellingShingle | Yasemen Ucan Resat Kosker The Real Forms of the Fractional Supergroup SL(2,C) Mathematics fractional supergroup Hopf algebra star-algebra |
| title | The Real Forms of the Fractional Supergroup SL(2,C) |
| title_full | The Real Forms of the Fractional Supergroup SL(2,C) |
| title_fullStr | The Real Forms of the Fractional Supergroup SL(2,C) |
| title_full_unstemmed | The Real Forms of the Fractional Supergroup SL(2,C) |
| title_short | The Real Forms of the Fractional Supergroup SL(2,C) |
| title_sort | real forms of the fractional supergroup sl 2 c |
| topic | fractional supergroup Hopf algebra star-algebra |
| url | https://www.mdpi.com/2227-7390/9/9/933 |
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