The three graces in the Tits--Kantor--Koecher category
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912339114065920 |
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| author | Dotsenko, Vladimir Kashuba, Iryna |
| author_facet | Dotsenko, Vladimir Kashuba, Iryna |
| contents | A metaphor of Loday describes Lie, associative, and commutative associative algebras as ``the three graces'' of the operad theory. In this article, we study the three graces in the category of $\mathfrak{sl}_2$-modules that are sums of copies of the trivial and the adjoint representation. That category is not symmetric monoidal, and so one cannot apply the wealth of results available for algebras over operads. Motivated by a recent conjecture of the second author and Mathieu, we embark on the exploration of the extent to which that category ``pretends'' to be symmetric monoidal. To that end, we examine various homological properties of free associative algebras and free associative commutative algebras, and study the Lie subalgebra generated by the generators of the free associative algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20635 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The three graces in the Tits--Kantor--Koecher category Dotsenko, Vladimir Kashuba, Iryna K-Theory and Homology Rings and Algebras A metaphor of Loday describes Lie, associative, and commutative associative algebras as ``the three graces'' of the operad theory. In this article, we study the three graces in the category of $\mathfrak{sl}_2$-modules that are sums of copies of the trivial and the adjoint representation. That category is not symmetric monoidal, and so one cannot apply the wealth of results available for algebras over operads. Motivated by a recent conjecture of the second author and Mathieu, we embark on the exploration of the extent to which that category ``pretends'' to be symmetric monoidal. To that end, we examine various homological properties of free associative algebras and free associative commutative algebras, and study the Lie subalgebra generated by the generators of the free associative algebra. |
| title | The three graces in the Tits--Kantor--Koecher category |
| topic | K-Theory and Homology Rings and Algebras |
| url | https://arxiv.org/abs/2310.20635 |