arXiv research

The paper studies curvatures in metric Jordan algebras, proving unique connections and curvature formulas.

We present new results about Jordan algebras and Jordan coalgebras, and we discuss about their connections with the Yang-Baxter equations.

Researchers solve the realization of Jordan-Kronecker invariants in Lie algebras.

Jordan algebras in information geometry linked to metrics on probability distributions.

We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use these examples to establish some results concerning higher order Osserman and highe…

The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments ar…

The paper classifies hypercomplex Lie algebras and solvmanifolds using quaternionic Jordan form.

Study on deformations of symmetric spaces using Jordan algebras.

Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.

The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characterist…

Let $R$ be an algebraic curvature tensor on a vector space of signature $(p,q)$ defining a spacelike Jordan Osserman Jacobi operator $\JJ_R$. We show that the eigenvalues of $\JJ_R$ are real and that $\JJ_R$ is diagonalizable if $p<q$.

Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-o…

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields $\K$, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generaliz…

We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic rational Cherednik algebra in the framed case. We also prove that the quanti…

Let $\mathcal{J}^1$ be the real form of complex simple Jordan algebra with the automorphism group $G$ of type $F_{4(-20)}$. Explicitly, we give the orbit decomposition of $\mathcal{J}^1$ under the action of $G$ and determine the Lie group structure of stabilizer for each $G$-orbit on $\mathcal{J}^1$.

We know that any element $X$ of the exceptional Jordan algebra $\gJ$ is transformed to a diagonal form by the compact exceptional Lie group $F_4$. However, its proof is used the method which is reduced a contradiction. In this paper, we give a direct and constructive proof.

We show that if $\nabla R$ is a Jordan Szabo algebraic covariant derivative curvature tensor on a vector space of signature (p,q), where q is odd and p is less than q or if q is congruent to 2 mod 4 and if p is less than q-1, then $\nabla R=0$. This algebraic result yields an elementary proof of the geometrical fact th…

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor $b_{ij}$ is named `compatible' with the curvature tensor if $b_i{}^m K_{jklm} + b_j{}^m K_{kilm} + b_k{}^m K_{ijlm} = 0$. Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetriz…

Let $\mathcal{J}^1$ be the real form of a complex simple Jordan algebra such that the automorphism group is $\mathrm{F}_{4(-20)}$. By using some orbit types of $\mathrm{F}_{4(-20)}$ on $\mathcal{J}^1$, for $\mathrm{F}_{4(-20)}$, explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's $K_ε-$Iwasawa decomp…

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …

A pseudo-Riemannian manifold is said to be spacelike Jordan IP if the Jordan normal form of the skew-symmetric curvature operator depends upon the point of the manifold, but not upon the particular spacelike 2-plane in the tangent bundle at that point. We use methods of algebraic topology to classify connected spacelik…

Let $U \subset \mathbb A^n$ be an open subset of real affine space. We consider functions $F: U \to \mathbb R$ with non-degenerate Hessian such that the first or the third derivative of $F$ is parallel with respect to the Levi-Civita connection defined by the Hessian metric $F"$. In the former case the solutions are gi…

We prove that for an algebraic curvature tensor on a pseudo-Euclidean space, the Jordan-Osserman condition implies the Rakić duality principle, and that the Osserman condition and the duality principle are equivalent in the diagonalisable case.

Classifies 2-solvable Frobenius Lie algebras based on endomorphisms.

We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ so that at each point of $M$ the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least …

In the algebraic context, we show that null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2,2). We also classify the null Jordan Osserman models of signature (2,2). In the geometric context, we show that a pseudo-Riemannian manifold of signature (2,2) is null Jo…

Relates two types of skein algebras using explicit correspondences.

New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.

The paper studies connections and Finsler geometry on JB-algebra structure groups.

We consider non-degenerate graph immersions into affine space $\mathbb A^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs $(J,γ)$, where $J$ is an $n$-dimensional real Jordan algebra and $γ$ is a no…

The paper proves that any smooth curve can have two similar inscribed rectangles.

Non-associtive algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we present a unification for associative algebras, Jordan algebras and Lie algebras…

The paper counts conjugacy classes of loxodromic elements in Anosov subgroups with a power saving error term.

We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.

Abstract classifies Lie algebras with complex or symplectic structures.

Normal forms and moduli stacks for flat connections on complex manifolds.

Study open orbits in causal flag manifolds with applications in AQFT.

I apply the algebraic classification of self-adjoint endomorphisms of ${\bf R}^{2,2}$ provided by their Jordan canonical form to the Ricci curvature tensor of four-dimensional neutral manifolds and relate this classification to an algebraic classification of the Ricci curvature spinor. These results parallel similar re…

Generic Hitchin representations avoid hyperplanes in Lie algebras.

A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric nilpotent groups. The nonsurjective injection defining such an extension induces…

Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $π$ is a spacelike 2 plane, let $R(π)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hyp…

The method of double extension, introduced by A.~Medina and Ph.~Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for instance for Lie superalgebras and associative algebras, Filippov $n$-algebras and Jord…

We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary generalizations, commutative associative and Jordan algebras with an invariant form. We…

Study correlations of spectral lengths and displacements in higher rank groups.

Paper defines multiplicities for quaternion eigenvalues without complex matrix concepts.

We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is desc…

Study pseudo-Riemannian metrics on Jordan superalgebras.