Malcev–Poisson–Jordan algebras

Abstract

Malcev–Poisson–Jordan algebra (MPJ-algebra) is defined to be a vector space endowed with a Malcev bracket and a Jordan structure which are satisfying the Leibniz rule. We describe such algebras in terms of a single bilinear operation, this class strictly contains alternative algebras. For a given Malcev algebra [Formula: see text], it is interesting to classify the Jordan structure ∘ on the underlying vector space of [Formula: see text] such that [Formula: see text] is an MPJ-algebra (∘ is called an MPJ-structure on Malcev algebra [Formula: see text]. In this paper we explicitly give all MPJ-structures on some interesting classes of Malcev algebras. Further, we introduce the concept of pseudo-Euclidean MPJ-algebras (PEMPJ-algebras) and we show how one can construct new interesting quadratic Lie algebras and pseudo-Euclidean Malcev (non-Lie) algebras from PEMPJ-algebras. Finally, we give inductive descriptions of nilpotent PEMPJ-algebras.

Split Malcev-Poisson-Jordan algebras

“…First two items can be proved as in the finite dimensional case (see [5] and references therein). The third can be verified by taking into account (1). In fact, for any h ∈ H, v α ∈ P α , and…”

“…Recently Malcev-Poisson-Jordan algebras were presented in [1] as a natural reformulation of Poisson algebras endowed with a Malcev structure and Jordan conditions (instead of the Lie structure and associativity, respectively) related by a Leibniz identity. Furthermore, in the referred paper the authors presented Malcev-Poisson-Jordan structures on some classes of Malcev algebras, also study the concept of pseudo-Euclidean Malcev-Poisson-Jordan algebras and nilpotent pseudo-Euclidean Malcev-Poisson-Jordan algebras are defined.…”

Splittings of operations for Lie-Poisson triple systems and related algebraic structures

“…Combining a L.t.s and a commutative associative algebraic structure into an appropriate way, we obtain the notion of Lie-Poisson triple system introduced in [2].…”

“…Proof. We know that every Malcev-Poisson algebra is a Malcev-Poisson-Jordan algebra (see [2]). The authors proved that the bracket…”