A unified approach to the definition of anti-algebras

Abstract

We introduce the $\mathfrak{M}$arseille anti-construction $\mathfrak{Ma}$ for varieties of algebras defined by multilinear identities of degree $3$, obtained by changing the sign of the right-nested part of each defining polynomial. This gives, for any variety $\Var$, a new variety $\mathfrak{Ma}$-$\Var$ that generalizes antiassociative algebras and the recently studied anti-Novikov and anti-right-symmetric $($pre-Jacobi--Jordan$)$ algebras. We describe free objects in several $\mathfrak{Ma}$-varieties. On the operadic level, over a field of characteristic $\mathrm{char}\mathbb{k}\ne 2$, we show $(\mathfrak{Ma}\text{-}\Var)^{(!)}=\mathfrak{Ma}\text{-}(\Var^{(!)})$ and that the Dong property is preserved by $\mathfrak{Ma}.$ Finally, we obtain the necessary and sufficient condition for $\mathfrak{Ma}\text{-}\Var=\Var$. The second part of the paper is dedicated to another type of anti-algebras, called $\mathcal{M}anin$ anti-algebras. We give the description of this type of algebras via calculation of the white Manin product of suitable operads and present various examples of known $\mathcal{M}anin$ anti-algebras. In the end, we proved that the symmetric affinization of an algebra gives a Jacobi-Jordan algebra if and only if the original algebra is a $\mathcal{M}a$-Novikov algebra, that is, an analog of a well-known result about affinization of Novikov algebras given by Balinskii and Novikov.