I shall report on a joint project with Iryna Kashuba, where we study commutative, associative, and Lie algebras in a category which is not symmetric monoidal (so that the corresponding categories of algebras are not categories of algebras over operads); we refer to that category as the Tits--Kantor--Koecher category since it essentially emerged in the works of these authors relating various questions on Jordan algebras to Lie theory. Recently, Iryna Kashuba and Olivier Mathieu formulated a conjecture about homology of free Lie algebras in that category, which, if true, would lead to a beautiful formula for dimensions of components of free Jordan algebras. Our work, originally motivated by that conjecture, does not resolve it, but unravels several somewhat surprising results that I shall present.
