The Freudenthal-Springer-Tits constructions revisited

It is well known that the Lie algebra of derivations of the exceptional Jordan algebra M 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional Lie algebra F 4 of Killing-Cartan and that the Lie algebra of linear transformations in M 3 8 leaving a certain cubic norm form defined on M 3 8 invariant is the exceptional Lie algebra E 6 (Chevalley-Schafer [7], Freudenthal [11]). The program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple Jordan algebras. The present paper gives the definition of the groups and Lie algebras in the general case and gives their determination for special Jordan algebras with some restrictions on the base field. In a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple Jordan algebras and in a third paper we hope to study the generalized groups of types E 6 which arise from these Jordan algebras.

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 exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space C⊕C3 is associated to the quark–lepton symmetry (one complex for the lepton and 3 for the corresponding quark). More generally it is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of “the algebra of real functions” on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.

The notions of the rank or principal polynomial of an associative algebra and the corresponding notions of trace and norm are classical. These notions have been generalized recently by the author ([13,I]) to apply to strictly power associative algebras, and we have renamed these concepts the generic minimal polynomial, trace and norm, since this terminology appears to be more in keeping with present day usage in analogous situations. In our paper we investigated the groups of linear transformations which preserve the norms in special central simple Jordan algebras. This applies to central simple associative algebras as a special case. In a later paper ([13, III]) we studied the norm preserving groups of exceptional central simple Jordan algebras. The groups obtained in this way are generalizations of the complex Lie group E 6 and include certain geometrically defined subgroups of the collineation groups of Cayley planes.

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra $\frak{J}$. The automorphism group $\rm{F}_4$ of $\frak{J}$ and its maximal Borel-de~Siebenthal subgroups $\frac{\rm SU(3)\times \rm SU(3)}{\mathbb{Z}_3}$ and ${\rm Spin}(9)$ are studied in some detail with an eye to possible applications to the fundamental fermions in the Standard Model of particle physics.

This result has a number of important consequences. It may be seen to imply that no simple exceptional finite dimensional Jordan algebra of characteristic not two is a homomorphic image of a special(2) Jordan algebra. But there is an exceptional simple Jordan algebra S& (of dimension 27) over a field a of characteristic not two which is generated by three of its elements. Then ID is a homomorphic image of the free Jordan algebra '3 on three generators. It follows that '3 is not special, and we also know that '3 is not isomorphic to the free special Jordan algebra 3 [x, y, z, 1 ] consisting of all reversible polynomials(') in the free associative algebra a [x, y, , 1Z] of all polynomials on the three generators x, y, z. 2. Elementary properties of 60 and certain subalgebras. We begin with a brief description of the algebra eo of our theorem. Let Z be an algebra with a unity element e over a field 0 of characteristic not two. Suppose that Z has an involution T over a; that is, an antiautomorphism

Commutativity preserving linear maps on central simple algebras

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal. Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra and provide a complete characterization of the theory of connections for free Jordan modules.

A well known theorem, due to Wedderburn, states that if $$ \mathfrak{B} $$ is an arbitrary associative algebra with an identity 1 and $$ \mathfrak{A} $$ is a finite dimensional central simple subalgebra containing 1, then $$ \mathfrak{B} $$ is the Kronecker product $$ \mathfrak{B} = \mathfrak{A} \otimes \mathfrak{U} $$ where $$ \mathfrak{U} $$ is the subalgebra of elements which commute with every $$ a\,\varepsilon \,\mathfrak{A} $$ . Recently Kaplansky indicated that the same result holds for $$ \mathfrak{B} $$ any alternative algebra and $$ \mathfrak{A} $$ an algebra of Cayley numbers.2 It is the purpose of the present note to give a new proof of this result and to show that an analogous result holds for $$ \mathfrak{B} $$ any Jordan algebra and $$ \mathfrak{A} $$ an exceptional simple Jordan algebra of finite dimensions.

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.

The simple root system of each exceptional, simple Lie algebra is explicitly constructed in a variety of forms. Each construction is based on the natural embedding in the exceptional algebra of a classical, semi-simple algebra of the same rank. The procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented Dynkin diagrams. The various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras. For each such labelling scheme the modification rules for dealing with non-standard representation labels are tabulated. The connection between the natural labels and Dynkin labels is given in detail and a comparison is made with the labels of B.G. Wybourne and M.J. Bowick (1977).

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.

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 $K_{\epsilon}-$Iwasawa decomposition, the Matsuki decomposition, and the Bruhat and Gauss decompositions.

We give an invariant classification of orbits of the fundamental representations of the exceptional groups E7(7) and E6(6) which classify BPS states in string and M theories toroidally compactified to d=4 and 5. The exceptional Jordan algebra and the exceptional Freudenthal triple system and their cubic and quartic invariants play a major role in this classification. The cubic and quartic invariants correspond to the black hole entropy in d=5 and d=4, respectively. The classification of BPS states preserving different numbers of supersymmetries is in close parallel to the classification of the little groups and the orbits of timelike, lightlike and spacelike vectors in Minkowski space. The orbits of BPS black holes in N=2 Maxwell–Einstein supergravity theories in d=4 and d=5 with symmetric space geometries are also classified including the exceptional N=2 theory, which has E7(-25) and E6(-26) as its symmetry in the respective dimensions.

While most theorists are tied to the mast of four dimensions, some have found it irresistible to speculate about eleven dimensions, the domain of M-theory. We outline a program which starts from the light-cone description of supergravity, and tracks its divergences to suggest the existence of an infinite component theory which in the light-cone relies on the coset $F_4/SO(9)$, long known to be linked to the Exceptional Jordan Algebra.