On the Definition of Quasi-jordan Algebra

Velásquez and Felipe recently introduced quasi-Jordan algebras based on the product a / b = 1 2 (a a b + b ` a) in an associative dialgebra with operations a and `. We determine the polynomial identities of degree ≤ 4 satisfied by this product. In addition to right commutativity and the right quasi-Jordan identity, we obtain a new associator-derivation identity. Loday [1, 2] defined an (associative) dialgebra to be a vector space with two bilinear operations a a b and a ` b satisfying these polynomial identities: (a a b) a c = a a (b a c), (a ` b) ` c = a ` (b ` c), (a ` b) a c = a ` (b a c), (a ` b) ` c = (a a b) ` c, a a (b a c) = a a (b ` c). Let w be a dialgebra monomial over X: w = a1 · · · an for a1, . . . , an ∈ X where the bar indicates some placement of parentheses and some choice of operations. We define the center c(w): if w ∈ X then c(w) = w; otherwise c(w1 a w2) = c(w1) and c(w1 ` w2) = c(w2). Lemma 1. [2] If w = a1 · · · an and c(w) = ak then w = (a1 ` · · · ` ak−1) ` ak a (ak+1 a · · · a an). We write w = a1 · · · ak−1âkak+1 · · · an for this normal form of w. Lemma 2. [2] The monomials a1 · · · ak−1âkak+1 · · · an (1 ≤ k ≤ n; a1, . . . , an ∈ X) form a basis of the free dialgebra on X. Velásquez and Felipe [3, 4] introduced the (right) quasi-Jordan product in a dialgebra over a field of characteristic 6= 2: a / b = 12 (a a b + b ` a). We omit the operation symbol and the scalar 12 , and write ab = a a b + b ` a. Lemma 3. [3] The quasi-Jordan product satisfies the right commutative identity: