Narrow Results

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 $(P,[,])$, it is interesting to classify the Jordan structure ∘ on the underlying vector space of $P$ such that $(P,[,],\circ )$ is an MPJ-algebra (∘ is called an MPJ-structure on Malcev algebra $(P,[,]))$. 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.

We study the variety of complex $n$-dimensional Jordan algebras using techniques from Geometric Invariant Theory. More specifically, we use the Kirwan–Ness theorem to construct a Morse-type stratification of the variety of Jordan algebras into finitely many invariant locally closed subsets, with respect to the energy functional associated to the canonical moment map. In particular we obtain a new, cohomology-free proof of the well-known rigidity of semisimple Jordan algebras in the context of the variety of Jordan algebras.

Let $𝔍$ and ${𝔍}^{\prime }$ be two ∗-Jordan algebras with identities $I_{𝔍}$ and $I_{{𝔍}^{\prime }}$, respectively, and $e$ a nontrivial ∗-idempotent in $𝔍$. In this paper, we study the characterization of multiplicative ∗-Jordan-type maps. In particular, we provide a characterization in the case of unital prime associative algebra endowed with an involution.