We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.

We begin by showing that isotopes of Albert algebras are obtained as twists by a certain $\mathrm F_4$ -torsor with total space a group of type $\mathrm E_6$ and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain $\mathrm D_4$ -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\,>\,1$ is a positive integer. We characterize $a\,\in \,K$, the Lie algebra of skew elements in $R$, satisfying ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,0$ on $K$. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if $a,\,b\,\in \,R$ satisfy ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,\text{a}{{\text{d}}_{b}}$ on $R$, where either $n$ is even or $b\,=\,0$, then ${{(a\,-\,\lambda )}^{[(n+1)/2]}}\,=\,0$ for some $\lambda \,\in \,C$.

We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $\left( V,\,Q \right)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi $ is an orbit of a covering of Conf$\left( V,\,Q \right)$, the conformal group of the pair $\left( V,\,Q \right)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak{g}$, and furthermore a real form ${{\mathfrak{g}}_{\mathbb{R}}}$. The connected and simply connected Lie group ${{G}_{\mathbb{R}}}$ with $\text{Lie}\left( {{G}_{\mathbb{R}}} \right)\,=\,{{\mathfrak{g}}_{\mathbb{R}}}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi $.

When it is usually using a bigger algebra system to formulate the Maxwell equations, in this paper we consider a real four-dimensional algebra to express the Maxwell equations without appealing to the imaginary number and higher dimensional algebras. In terms of g-based Jordan algebra formulation the Lorentz gauge condition is found to be a necessary and sufficient condition to render the second pair of Maxwell equations, while the first pair of Maxwell equations is proved to be an intrinsic algebraic property. Then, we transform the g-based Jordan algebra to a Lie algebra of the dilation proper orthochronous Lorentz group, which gives us an incentive to consider a linear matrix operator of the Lie type, rendering more easy to derive the Maxwell equations and the wave equations. The new formulations fully match the requirements for the classical electrodynamic equations and the Lorentz gauge condition. The mathematical advantage of our formulations is that they are irreducible in the sense that, when compared to the formulations which using other bigger algebras (e.g., biquaternions and Clifford algebras), the number of explicit components and operations is minimal. From this aspect, the g-based Jordan algebra and Lie algebra are the most suitable algebraic systems to implement the Maxwell equations into a more compact form.

We obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time $t$ and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.