Exceptional Simple Jordan Algebras Research Articles

Gradings on the exceptional Lie algebras $F_4$ and $G_2$ revisited

All gradings by abelian groups are classified on the following algebras over an algebraically closed field \mathbb{F} : the simple Lie algebra of type G_2 (char \mathbb{F}\ne 2,3 ), the exceptional simple Jordan algebra (char \mathbb{F}\ne 2 ), and the simple Lie algebra of type F_4 (char \mathbb{F} \ne 2 ).

Springer forms and the first Tits construction of exceptional Jordan division algebras

In this paper, a certain quadratic form, originally due to Springer [15], which may be associated with any separable cubic subfield living inside an exceptional simple Jordan algebra is related to the coordinate algebra of an appropriate scalar extension. We use this relation to show that, in the presence of the third roots of unity, exceptional Jordan division algebras arising from the first Tits construction are precisely those where reducing fields and splitting fields agree, and that all isotopes of a first construction exceptional division algebra are isomorphic.

Composition series and intertwining operators for the spherical principal series. II

In this paper, we consider the connected split rank one Lie group of real type F 4 {F_4} which we denote by F 4 1 F_4^1 . We first exhibit F 4 1 F_4^1 as a group of operators on the complexification of A. A. Albert’s exceptional simple Jordan algebra. This enables us to explicitly realize the symmetric space F 4 1 / Spin ( 9 ) F_4^1/{\text {Spin}}(9) as the unit ball in R 16 {{\mathbf {R}}^{16}} with boundary S 15 {S^{15}} . After decomposing the space of spherical harmonics under the action of Spin ( 9 ) {\text {Spin}}(9) , we obtain the matrix of a transvection operator of F 4 1 /Spin ( 9 ) F_4^1{\text {/Spin}}(9) acting on a spherical principal series representation. We are then able to completely determine the Jordan Holder series of any spherical principal series representation of F 4 1 F_4^1 .

Orbits of the automorphism group of the exceptional Jordan algebra.

Necessary and sufficient conditions for two elements of a reduced exceptional simple Jordan algebra a to be conjugate under the automorphism group Aut a of a are obtained. It was known previously that if :a is split, then such elements are exactly those with the same minimum polynomial and same generic minimum polynomial. Also, it was known that two primitive idempotents are conjugate under Aut a if and only if they have the same norm class. In the present paper the notion of norm class is extended and combined with the above conditions on the minimum and generic minimum polynomials to obtain the desired conditions for arbitrary elements of :a. In this paper we characterize the orbits of the automorphism group Aut a of a reduced exceptional simple Jordan algebra I = '(03, y) over a field 'D of characteristic not two or three. Since Jacobson [2, Theorem 9] has shown that in case the octonion (Cayley-Dickson) algebra D is split then such an orbit consists of exactly those elements with the same minimum polynomial and the same generic minimum polynomial, we shall restrict our attention to octonion division algebras. In particular, we shall assume 'I' is infinite. Recall that D(03, y) is the Jordan algebra of 3 x 3 symmetric matrices with entries in D with respect to the involution x I> y 1cty, y = diag {Yl, Y2, Y3}, 0 = Yi e (D, and with multiplication x y= I(xy +yx), xy the usual matrix multiplication. An element x E D(03, y) iS of the form (1) x = aieii aik] with aoi E , ai E , i = 1, 2, 3, where (i, j, k) is a cyclic permutation of (1, 2, 3) and a[ij] = yjaeij + yiceji. We have the mapping x i-x# defined by (2) x# = 2 (aak yykn(a?))eii + 2 (y1(ajak) -aia1)[jk] where x is as in (1) and n is the norm on D. If x#& 0 and x# = 0, we say x is of rank one. Also, we write xxy=(x+y)#-x#-y#. For x, ye 3 write T(x,y)=T(x.y) where T(z) is the usual trace. Received by the editors January 27, 1970. AMS 1969 Subject Classifications. Primary 1740.

The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras

Constructions of 27-dimensional exceptional simple Jordan algebras have been given by H. Freudenthal [2], [3], T. A. Springer [8], and J. Tits [9]. In the first two approaches the cubic generic norm form plays a central role, with applications to projective geometry and algebraic groups; the third approach gives a simple method for constructing all exceptional simple algebras. The constructions are limited to fields of characteristic #2 as usual for Jordan algebras defined in terms of a bilinear multiplication, and in order to polarize the cubic norm form the characteristic must be # 3. Recently a definition of Jordan algebras has been proposed [5] which is based on a cubic composition involving U-operators. A unital Jordan algebra over a commutative associative ring &P is a triple : = (X, U, 1) where X is a &P-module, U a quadratic mapping x -> Ux of X into Homo (X, X), and 1 an element of X satisfying the axioms

The classification of reduced exceptional simple jordan algebras

The classification of reduced exceptional simple jordan algebras

On a homomorphism property of certain Jordan algebras

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

A Construction of Exceptional Jordan Division Algebras

Every simple Jordan algebra ,) over a field H of characteristic not two has a unity element 1. Write 1 = e, + *-+ et for pairwise orthogonal primitive idempotents e,, and let 4? be the set of all elements xf of 1? such that em xi xi, where ab is the product in '?. Then g is said to be reduced of degree t if the t subalgebras As all have dimension one over A. If W is any associative algebra, and ab is the associative product of a, there is an attached Jordan algebra A'+) which consists of the vector space 2 and the product ab defined by 2ab = ab + ba. A Jordan algebra ? is said to be exceptional if ? is not isomorphic to a subalgebra of any W for W associative. It is known2 that the dimension of any simple exceptional Jordan algebra 1? over its center g is 27, and that there exists a scalar extension A, of finite degree over A, such that & is isomorphic to the Jordan algebra &3((S) of all three-rowed Hermitian matrices with elements in the split Cayley algebra (E of dimension 8 over R. Represent the product operation in the algebra (2, of all three-rowed matrices with elements in (E by ab, and the operation ab in 5?3(() is defined by 2ab ab + ba. In 1948 R. D. Schafer stated3 that every exceptional simple Jordan algebra is reduced. His proof of the statement contains an error4, and the result is false. We shall construct a class of central simple exceptional Jordan algebras ? over any field g of cearacteristic not two which we shall call cyclic Jordan algebras. Each such algebra & has an attached cyclic associative algebra = (Sty Sy r) of degree three over its center A, and an attached Cayley algebra (E over St having a nonsingular linear transformation T over g inducing S in St and such that

On Reduced Exceptional Simple Jordan Algebras

On Reduced Exceptional Simple Jordan Algebras

A Kronecker Factorization Theorem for Cayley Algebras and the Exceptional Simple Jordan Algebra

A Kronecker Factorization Theorem for Cayley Algebras and the Exceptional Simple Jordan Algebra

Lie algebras of type 𝐹

Chevalley and Schafer [4 ]2 have shown that the exceptional simple Lie algebra F4 of dimension 52 over an arbitrary algebraically closed field Q2 of characteristic 0 is the derivation algebra of the unique exceptional simple Jordan algebra of dimension 27 over Q2. In this paper we show that a Lie algebra 2 over an arbitrary field 1D of characteristic 0 is of type F if and only if 2 is isomorphic to the derivation algebra Z(3) of an exceptional central simple Jordan algebra a over (D. The proof given for this theorem requires a characterization of the automorphisms of ZQ() over R. We prove that every automorphism of Z(a) has the form D-+SDS-1 for a unique automorphism S of a. The classification of Lie algebras of type F over 4) is reduced to the problem of classifying exceptional central simple Jordan algebras over (D, since it is shown that Z ,1)Z)($a2) if and only if 2 In the last section of this paper the three exceptional central simple Jordan algebras over a real closed field are exhibited and their derivation algebras are the real closed Lie algebras of type F.