Finite-dimensional Jordan algebras admitting the structure of a Jordan bialgebra

Special and exceptional mock-Lie algebras

“…And here comes another obstacle: in general, there is no notion of the tensor product of two representations of a mock-Lie algebra. Perhaps, one may try to work around it by defining an ad-hoc bialgebra structure on mock-Lie algebras in question using central elements, similarly how it is done in [Zhel,Propositions 1 and 2] in some particular cases of Jordan algebras, and get in this way that any mock-Lie algebra of nilpotency index ≤ 4 has a faithful representation, and hence is special. This result, however, would be covered by Theorem 2, or, more generally, by the above-cited Slin'ko's result about speciality of nilpotent Jordan algebras of nilpotency index ≤ 5.…”

A double construction of quadratic anticenter-symmetric Jacobi-Jordan algebras

“…This class of algebras appear under different names in the litterature reflecting, perhaps, the fact that it was considered from different viewpoints by different communities, sometimes not aware of each other's results. In [11,13,10] and other Jordan litterature, these algebras are called Jordan algebras of nil index 3. In [9] they are called Lie-Jordan algebras (superalgebras are also considered there).…”

On a pre-Jacobi-Jordan algebra: relevant properties and double construction

“…Apparently this class of algebras appear under different names in the litterature reflecting, perhaps, the fact that it was considered from different viewpoints by different communities, sometimes not aware of each other's results. In [6][7][8] and other Jordan litterature,these algebras are called Jordan algebras of nil index 3. In [10] they are called Lie-Jordan algebras (superalgebras are also considered there).…”