Exceptional Jordan Algebra Research Articles

A note on reduced Jordan algebras

In this paper we give short proofs of two of the main theorems concerning reduced exceptional Jordan algebras A = H(C3, 'y): the AlbertJacobson Theorem that the Cayley coordinate algebra C is determined by A up to isomorphism, and the Springer Theorem that two such algebras A, A' are isomorphic if and only if they have isomorphic coordinate algebras and equivalent trace forms. We avoid using the generic norm by working directly with the reduced idempotents. Our proofs do not require that the algebras be exceptional, and are valid for arbitrary reduced simple algebras. 1. Reduced idempotents. Throughout this paper, A will denote a Jordan algebra with identity element 1 over a field 1 of characteristic 5 2. In this section we make no assumptions about the simplicity or finite-dimensionality of A. Recall that an idempotent e is reduced if Ai(e) = UeA =cJe. We assume the reader is familiar with the operators U, = 2L' -Lx2and the basic identities involving them (e.g. [3,

Simple Algebras that Generalize the Jordan Algebra M38

In this paper we discuss a generalization of the split exceptional Jordan algebra M38() of the 3 X 3 hermitian matrices with elements in the split Cayley-Dickson algebra (1). The generalization consists of replacing by the non-commutative Jordan algebra ≡ (A, ƒ, s, t) discussed in (2; 3) and forming the set of 3 X 3 hermitian matrices M3m() ≡ M with elements in the m-dimensional algebra . With the usual definition of multiplication X · Y = ½(XY + YX), M becomes a commutative algebra and we have the following theorem, which shows how the structure of M is reflected by that of .

On automorphisms of Lie algebras of classical type. III

ceptional Jordan algebra a. Except for fields of characteristic 2 and 3, the interpretations of these Lie algebras given for characteristic zero by Chevalley and Schafer [3 ] are shown to carry over to arbitrary base fields. When several more low characteristics are exempted from consideration (on the basis of a failure of applicability of certain results on exponentials in [10]), the structure of the automorphism groups of these algebras is here determined in terms of certain linear groups acting in a. Some of the latter groups have also been studied recently by Jacobson [6; 7], who has proved the simplicity of a group here shown to be isomorphic to the group of invariant automorphisms of the Lie algebra in question (F4 or E6). The present paper also demonstrates the simplicity of these groups, by showing that they can be identified with groups whose simplicity has been proved by Chevalley [2]. 1. Derivations of the split exceptional Jordan algebra. Let (? be the split Cayley algebra over a field j of characteristic F 2, regarded as all matrices of the form ta

On automorphisms of Lie algebras of classical type. II

ceptional Jordan algebra a. Except for fields of characteristic 2 and 3, the interpretations of these Lie algebras given for characteristic zero by Chevalley and Schafer [3 ] are shown to carry over to arbitrary base fields. When several more low characteristics are exempted from consideration (on the basis of a failure of applicability of certain results on exponentials in [10]), the structure of the automorphism groups of these algebras is here determined in terms of certain linear groups acting in a. Some of the latter groups have also been studied recently by Jacobson [6; 7], who has proved the simplicity of a group here shown to be isomorphic to the group of invariant automorphisms of the Lie algebra in question (F4 or E6). The present paper also demonstrates the simplicity of these groups, by showing that they can be identified with groups whose simplicity has been proved by Chevalley [2]. 1. Derivations of the split exceptional Jordan algebra. Let (? be the split Cayley algebra over a field j of characteristic F 2, regarded as all matrices of the form ta\

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.