Graded Medial <i>n</i>-Ary Algebras and Polyadic Tensor Categories
Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or <inline-formula><math xmlns="http://www.w3.org/1998/Math/Math...
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MDPI AG
2021-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/6/1038 |
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| author | Steven Duplij |
| author_facet | Steven Duplij |
| author_sort | Steven Duplij |
| collection | DOAJ |
| description | Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an <i>n</i>-ary tensor product as an additional multiplication with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> associators of the arity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> satisfying a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mfenced></semantics></math></inline-formula>-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in <i>n</i>-ary groups). The arity-nonreducible <i>n</i>-ary braiding is introduced and the equation for it is derived, which for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories. |
| first_indexed | 2024-03-10T10:35:28Z |
| format | Article |
| id | doaj.art-b905b171085f4439b01f8b11b20f4082 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T10:35:28Z |
| publishDate | 2021-06-01 |
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| series | Symmetry |
| spelling | doaj.art-b905b171085f4439b01f8b11b20f40822023-11-21T23:20:21ZengMDPI AGSymmetry2073-89942021-06-01136103810.3390/sym13061038Graded Medial <i>n</i>-Ary Algebras and Polyadic Tensor CategoriesSteven Duplij0Center for Information Technology (WWU IT), Universität Münster, Röntgenstrasse 7-13, 48149 Münster, GermanyAlgebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an <i>n</i>-ary tensor product as an additional multiplication with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> associators of the arity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> satisfying a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mfenced></semantics></math></inline-formula>-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in <i>n</i>-ary groups). The arity-nonreducible <i>n</i>-ary braiding is introduced and the equation for it is derived, which for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.https://www.mdpi.com/2073-8994/13/6/1038gradingcommutativitymedialitytensor categorymonoidal categorybraiding |
| spellingShingle | Steven Duplij Graded Medial <i>n</i>-Ary Algebras and Polyadic Tensor Categories Symmetry grading commutativity mediality tensor category monoidal category braiding |
| title | Graded Medial <i>n</i>-Ary Algebras and Polyadic Tensor Categories |
| title_full | Graded Medial <i>n</i>-Ary Algebras and Polyadic Tensor Categories |
| title_fullStr | Graded Medial <i>n</i>-Ary Algebras and Polyadic Tensor Categories |
| title_full_unstemmed | Graded Medial <i>n</i>-Ary Algebras and Polyadic Tensor Categories |
| title_short | Graded Medial <i>n</i>-Ary Algebras and Polyadic Tensor Categories |
| title_sort | graded medial i n i ary algebras and polyadic tensor categories |
| topic | grading commutativity mediality tensor category monoidal category braiding |
| url | https://www.mdpi.com/2073-8994/13/6/1038 |
| work_keys_str_mv | AT stevenduplij gradedmedialiniaryalgebrasandpolyadictensorcategories |