Related papers: Primitive 4-generated axial algebras of Jordan typ…
We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $1$ or $3$. We prove that the variety is the union of Zariski closures of the orbits of $11$ and $21$ rigid superalgebras,…
We study the variety of complex $n$-dimensional Jordan algebras using techniques from Geometric Invariant Theory.
We show that a primitive axial algebra of Jordan type $\eta = \tfrac{1}{2}$ is a Jordan algebra if and only if every $2$-generated subalgebra is \emph{solid}, a notion introduced recently by Ilya Gorshkov, Sergey Shpectorov and Alexei…
We introduce and investigate new classes of Jordan algebras which are close to but wider than Rickart and Baer Jordan algebras considered in our previous paper. Such Jordan algebras are called RJ- and BJ-algebras respectively. Criterions…
We show that Artin-Schelter regularity of a $\mathbb{Z}$-graded algebra can be examined by its associated $\mathbb{Z}^r$-graded algebra. We prove that there is exactly one class of four-dimensional Artin-Schelter regular algebras with two…
The Jordan algebra of the symmetric matrices of order two over a field $K$ has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is…
In this paper, we compute all possible Jordan types of linear forms $\ell$ in any full Perazzo algebra $A$. In some cases we are also able to compute the corresponding Jordan degree type, which is a finer invariant.
In this paper we determine all types and the canonical forms of simple subalgebras for each type of simple Jordan algebras and the number of conjugate classes corresponding to the given simple Jordan algebra.
We discuss whether the Jordan degree type encodes \break more information about graded artinian Gorenstein algebras than the Jordan type for linear forms. We show that in codimension two, the Jordan type determines the Jordan degree type.…
Abelian varieties of dimension 2n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n(n-1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomology group. In…
The class of algebras of Jordan type $\eta$ was introduced by Hall, Rehren and Shpectorov in 2015 within the much broader class of axial algebras. Algebras of Jordan type are commutative algebras $A$ over a field of characteristic not $2$,…
In this paper, we develop a method to obtain the algebraic classification of noncommutative Jordan algebras from the classification of Jordan algebras of the same dimension. We use this method to obtain the algebraic classification of…
In this work, the automorphism group schemes of finite-dimensional simple Jordan pairs of types I and IV, and of some Jordan triple systems related to them, are determined. We assume $\mathrm{char}(\mathbb{F}) \neq 2$ for the base field…
Jordan, Wigner and von Neumann classified the possible algebras of quantum mechanical observables, and found they fell into 4 "ordinary" families, plus one remarkable outlier: the exceptional Jordan algebra. We point out an intriguing…
We describe degenerations of three-dimensional Jordan superalgebras over $\mathbb{C}$. In particular, we describe all irreducible components in the corresponding varieties.
We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian.
We developed a new proper method for classifying $n$-dimensional derived Jordan algebras, and apply it to the classification of $3$-dimensional derived Jordan algebras. As a byproduct, we have the algebraic classification of $3$-dimensional…
Associative or Jordan algebras generated by two idempotents are described precisely.
A Jordan algebra J is said to be pseudo-euclidean if J is endowed with an associative non-degenerate symmetric bilinear form B. B is said an associative scalar product on J. First, we provide a description of the pseudo-euclidean Jordan…
First, we study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra. We give an isomorphic characterization of 2-step nilpotent pseudo-Euclidean Jordan algebras. Next, we…