Narrow Results

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.

A set of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are:

(1) Every irreducible semitransitive Jordan algebra is actually transitive.

(2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

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.

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.

We describe some natural relations connecting contact geometry, classical Monge–Ampère equations (MAEs) and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of MAEs and sheds new light on some aspects of contact geometry.

It is shown that the algebra $J_3[C\otimes O]\otimes Cl(4,C)$ based on the complexified Exceptional Jordan, and the complex Clifford algebra in 4D, is rich enough to describe all the spinorial degrees of freedom of three generations of fermions in 4D, and include additional fermionic dark matter candidates. Furthermore, the model described in this paper can account also for the Standard Model gauge symmetries. We extend these results to the Magic Star algebras of Exceptional Periodicity developed by Marrani–Rios–Truini and based on the Vinberg cubic $T$ algebras which are generalizations of exceptional Jordan algebras. It is found that there is a one-to-one correspondence among the real spinorial degrees of freedom of four generations of fermions in 4D with the off-diagonal entries of the spinorial elements of the $pair$$T_3^{8,n},({\bar{T}}_3^{8,n})$ of Vinberg matrices at level $n=2$. These results can be generalized to higher levels $n>2$ leading to a higher number of generations beyond 4. Three $pairs$ of $T$ algebras and their conjugates $\bar{T}$ were essential in the Magic Star construction of Exceptional Periodicity that extends the $e_8$ algebra to $e_8^{(n)}$ with $n$ integer.

Perfect C^∗-algebras were introduced by Akeman and Shultz in [Perfect C*-algebras, Mem. Amer. Math. Soc. 55(326) (1985)] and they form a certain subclass of C*-algebras determined by their pure states, and for which the general Stone–Weierstrass conjecture is true. In this paper, we introduce the notion of perfect JC-algebras, and we use the strong relationship between a JC-algebra A and its universal enveloping C^∗-algebra $C^{\ast }(A)$, to establish that if $C^{\ast }(A)$ is perfect and A is of complex type, then A is perfect. It is also shown that every scattered JC-algebra of complex type is perfect, and the same conclusion holds for every JC-algebra of complex type whose primitive spectrum is Hausdorff.