From Hopf Algebra to Braided <i>L</i><sub>∞</sub>-Algebra
We show that an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra can be extended to a...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/8/4/222 |
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| author | Clay James Grewcoe Larisa Jonke Toni Kodžoman George Manolakos |
| author_facet | Clay James Grewcoe Larisa Jonke Toni Kodžoman George Manolakos |
| author_sort | Clay James Grewcoe |
| collection | DOAJ |
| description | We show that an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra can be extended to a graded Hopf algebra with a codifferential. Then, we twist this extended <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra with a Drinfel’d twist, simultaneously twisting its modules. Taking the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra as its own (Hopf) module, we obtain the recently proposed braided <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra. The Hopf algebra morphisms are identified with the strict <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-morphisms, whereas the braided <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-morphisms define a more general <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-action of twisted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebras. |
| first_indexed | 2024-03-09T04:09:41Z |
| format | Article |
| id | doaj.art-726a097037c443d999587445424e2b93 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-09T04:09:41Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-726a097037c443d999587445424e2b932023-12-03T14:02:22ZengMDPI AGUniverse2218-19972022-04-018422210.3390/universe8040222From Hopf Algebra to Braided <i>L</i><sub>∞</sub>-AlgebraClay James Grewcoe0Larisa Jonke1Toni Kodžoman2George Manolakos3Division of Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, 10000 Zagreb, CroatiaDivision of Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, 10000 Zagreb, CroatiaDivision of Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, 10000 Zagreb, CroatiaDivision of Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, 10000 Zagreb, CroatiaWe show that an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra can be extended to a graded Hopf algebra with a codifferential. Then, we twist this extended <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra with a Drinfel’d twist, simultaneously twisting its modules. Taking the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra as its own (Hopf) module, we obtain the recently proposed braided <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra. The Hopf algebra morphisms are identified with the strict <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-morphisms, whereas the braided <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-morphisms define a more general <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-action of twisted <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebras.https://www.mdpi.com/2218-1997/8/4/222L∞-algebraHopf algebraDrinfel’d twist |
| spellingShingle | Clay James Grewcoe Larisa Jonke Toni Kodžoman George Manolakos From Hopf Algebra to Braided <i>L</i><sub>∞</sub>-Algebra Universe L∞-algebra Hopf algebra Drinfel’d twist |
| title | From Hopf Algebra to Braided <i>L</i><sub>∞</sub>-Algebra |
| title_full | From Hopf Algebra to Braided <i>L</i><sub>∞</sub>-Algebra |
| title_fullStr | From Hopf Algebra to Braided <i>L</i><sub>∞</sub>-Algebra |
| title_full_unstemmed | From Hopf Algebra to Braided <i>L</i><sub>∞</sub>-Algebra |
| title_short | From Hopf Algebra to Braided <i>L</i><sub>∞</sub>-Algebra |
| title_sort | from hopf algebra to braided i l i sub ∞ sub algebra |
| topic | L∞-algebra Hopf algebra Drinfel’d twist |
| url | https://www.mdpi.com/2218-1997/8/4/222 |
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