The three graces in the Tits--Kantor--Koecher category

Autor:	Dotsenko, Vladimir, Kashuba, Iryna
Rok vydání:	2023
Předmět:	Mathematics - K-Theory and Homology Mathematics - Rings and Algebras
Druh dokumentu:	Working Paper
Popis:	A metaphor of Jean-Louis 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. Comment: 17 pages, comments are welcome
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2310.20635 Zobrazit plný text záznamu View this record from Arxiv