Jacobi–Jordan algebras

On Bernstein algebras satisfying chain conditions II

“…It is well known that they are Jordan algebras (see [3, Lemma 1], [6, Lemma 2.2], [10], [30, page 114]). They appear in the litterature as Jacobi-Jordan algebras in [1,6] and as mock-Lie algebras in [22]. In addition, any such an algebra N is solvable and N (4) = 0 [31, Lemma 3.1].…”

Special and exceptional mock-Lie algebras

“…Conversely, linearizing the latter identity, we get back the Jacobi identity. Moreover, it is easy to see that assuming commutativity, the Jacobi identity is equivalent to the Jordan identity (x •2 • y) • x = x •2 • (y • x) (see, for example, [BF,Lemma 2.2]). (On the other hand, commutative Leibniz and commutative Zinbiel algebras form a narrower class of mock-Lie algebras, namely, commutative algebras of nilpotency index 3: (x • y) • z = 0).…”

“…Algebra from heading (i) is denoted as A 12 in this classification, and is the only nonabelian 2-dimensional algebra. Algebra from heading (ii), the "commutative Heisenberg algebra", is isomorphic to the algebra A 13 (see [BF,Remark 3.3]), and algebra from heading (iii) is isomorphic to A 12 ⊕ A 01 , the direct sum of algebra from heading (i) and one-dimensional abelian algebra. First proof of Theorem 1.…”

“…The table contains all nonabelian algebras of dimension ≤ 5, and all nonassociative algebras of dimension 6. We follow the nomenclature of [BF], with the exception of L which denotes the unique, over an algebraically closed field, 5-dimensional nonassociative mock-Lie algebra (see [BF,Proposition 4.1]). In the family of 6-dimensional algebras A 26 (β , 0) and A 26 (β , 1), the parameter β assumes values 0, 1.…”

On Jordan-Nilalgebras of Index 3

“…In [3], these algebras are called Jacobi-Jordan algebras, and the authors give a classification for dimension < 7 over algebraically closed fields of characteristics not 2 or 3. These algebras are also related to Bernstein-Jordan algebras and train algebras of rank 3.…”