Related papers: Exceptional symmetric domains
A simple unifying view of the exceptional Lie algebras is presented. The underlying Jordan pair content and role are exhibited. Each algebra contains three Jordan pairs sharing the same Lie algebra of automorphisms and the same external…
In this paper, we classify Jordan superalgebras of dimension up to three over an algebraically closed field of characteristic different of two. Our main motivation to obtain such classification comes out from the intention to give an answer…
We study isometric maps between Teichm\"uller spaces and bounded symmetric domains in their intrinsic Kobayashi metric. From a complex analytic perspective, these two important classes of geometric spaces have several features in common but…
In this thesis we study algebraic structures in M-theory, in particular the exceptional Lie algebras arising in dimensional reduction of its low energy limit, eleven-dimensional supergravity. We focus on e8 and its infinite-dimensional…
We review various properties of the exceptional Euclidean Jordan algebra of degree three. Euclidean Jordan algebras of degree three and their corresponding Freudenthal triple systems were recently shown to be intimately related to extremal…
In this paper we look into the structure of finite-dimensional graded superalgebras of various types such as associative, Lie and Jordan over an algebraically closed field of characteristic zero.
Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group $SU(3)$ and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to…
The maximal subalgebras of the finite dimensional simple special Jordan superalgebras over an algebraically closed field of characteristic 0 are studied. This is a continuation of a previous paper by the same authors about maximal…
We review exceptional field theories as the duality-covariant reformulation of maximal supergravity theories in ten and eleven dimensions, that make the underlying exceptional symmetries explicit. Beyond their structural role in unifying…
We present a periodic infinite chain of finite generalisations of the exceptional structures, including the exceptional Lie algebra $\mathbf{e_8}$, the exceptional Jordan algebra (and pair) and the octonions. We will also argue on the…
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.…
A representation of the exceptional Lie algebras is presented. It reflects a simple unifying view and it is realized in terms of Zorn-type matrices. The role of the underlying Jordan pair and Jordan algebra content is crucial in the…
In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils, Lauve and Witherspoon and we call the restricted Jordan plane. In this paper the characteristic is odd. The defining…
We give a classification up to equivalence of the fine group gradings by abelian groups on the Jordan pairs and triple systems of types bi-Cayley and Albert, under the assumption that the base field is algebraically closed of characteristic…
We give the algebraic and geometric classification of complex four-dimensional Jordan superalgebras. In particular, we describe all irreducible components in the corresponding varieties.
We solve an open problem in spectral geometry: the construction of finite-dimensional, discrete geometries coordinatized by non-simple, exceptional Jordan algebras. The approach taken is readily generalisable to broad classes of…
While describing the results of our recent work on exceptional Lie and Jordan algebras, so tightly intertwined in their connection with elementary particles, we will try to stimulate a critical discussion on the nature of spacetime and…
The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group of degree 4 on this…
We classify decompositions of simple special finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic zero into the sum of two proper simple subsuperalgebras.
These notes were written following lectures I had the pleasure of giving on this subject at Keio University, during November and December 2004. The first part is about new applications of Jordan algebras to the geometry of Hermitian…