Let # be the exceptional 27-dimensional Jordan algebra over C. Its automorphism group is the Lie group F4(@) and this group is known to have a finite subgroup AL, where A is a self centralizing elementary abelian of order 27, L g SL(3, 3), and L normalizes A. As an A-module, 2 decomposes into a direct sum of l-dimensional spaces jrwhich afford the 27 distinct linear characters x E A n := Hom(A,...

We describe all degenerations of the variety $\mathfrak{Jord}_3$ Jordan algebras dimension three over $\mathbb{C}.$ In particular, we irreducible components in $\mathfrak{Jord}_3.$ For every $n$ define an $n$-dimensional rigid ''marginal'' algebra level one. Also, discuss associative, alternative, left non-commutative Jordan, Leibniz, and anticommutative cases.

Here we demonstrate the emergence of Grassmann variables in matrix models based on the exceptional Jordan algebra. The Grassmann algebras are built naturally using the octonion algebra. We argue the appearance of Grassmann variables solidifies the relationship between supersymmetry and triality.

— If the space Q of quadratic forms in Rn is splitted in a direct sum Q1 ⊕ · · · ⊕ Qk and if X and Y are independent random variables of Rn, assume that there exist a real number a such that E(X|X + Y ) = a(X + Y ) and real distinct numbers b1, ..., bk such that E(q(X)|X + Y ) = biq(X + Y ) for any q in Qi. We prove that this happens only when k = 2, when Rn can be structured in a Euclidean Jor...

This paper describes an intuitive geometric model for coupled twostate quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geometric algebra, and the insights this ...

In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of ...

We identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold.

A supermanifold, Mm/n, can be caracterired by its smooth superfunctions which constitute an algebra A (Leites, Kostant). We associate canonically ~a Ia Gelfand, certain fibred manifolds on which the automorphisms (the Jordan automorphisms) of A actas diffeomorphisms. For example, the kernelsof all homomorphisms from the algebra of superfunctions onto the Grassmann algebra of dimension n form na...