Central Simple Jordan Algebras Research Articles

Differentiably simple Jordan algebras

We prove that each exceptional differentiably simple Jordan algebra over a field of characteristic 0 is an Albert ring whose elements satisfy a cubic equation with the coefficients in the center of the algebra. If the characteristic of the field is greater than 2 then such an algebra is the tensor product of its center and a central exceptional simple 27-dimensional Jordan algebra. Some remarks made on special algebras.

Cohomological invariants of Jordan algebras with frames

In a previous paper the author determined all possible cohomological invariants of Aut ( J ) -torsors in Galois cohomology with mod 2 coefficients, where J is a split central simple Jordan algebra of odd degree n ⩾ 3 . In the present paper we extend these results to a larger class of groups G n 1 , … , n d r indexed by r = 0 , 1, 2 or 3, and n i positive integers that add to n ⩾ 3 such that n 1 is odd. We also give precise values for the essential dimension of these groups.

Cohomological invariants of odd degree Jordan algebras

AbstractIn this paper we determine all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for J a split central simple Jordan algebra of odd degree n ≥ 3. This has already been done for J of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed(PSp2n) = n + 1 for n odd.

Homogeneous spaces with invariant projectively flat affine connections

We characterize invariant projectively flat affine connections in terms of affine representations of Lie algebras, and show that a homogeneous space admits an invariant projectively flat affine connection if and only if it has an equivariant centro-affine immersion. We give a correspondence between semi-simple symmetric spaces with invariant projectively flat affine connect;ions and central-simple Jordan algebras.

Tits‘ Constructions of Jordan Algebras and F4 Bundles on the Plane

In this paper, constructions of Jordan algebras over commutative rings are given which place, within a general set-up, the classical Tits constructions of exceptional central simple Jordan algebras over fields. These are used to exhibit nontrivial Jordan algebra bundles over the affine plane with a given exceptional Jordan division algebra over k as the fibre. The associated principal F4 bundles are shown to admit no reduction of the structure group to any proper connected reductive subgroup.

On algebras of rank three

An algebra of rank three is a commutative, finite dimensional algebra that may be defined by the property that every element generates a subalgebra of dimension not greater than two. In this article we discuss several classes of such algebras, including two classes related to central simple Jordan algebras, and derive some general results which indicate that, with the exception of one pathological class related to nilpotent algebras, every rank three algebra can be constructed either from a quadratic and alternative algebra or from a representation of a Clifford algebra. Among other results, semisimple and simple rank three algebras are characterized, and the radical of an arbitrary rank three algebra is determined.

On the Invariants mod 2 of Albert Algebras

The existence of a new invariant for central simple exceptional Jordan algebras (nowadays called Albert algebras) over an arbitrary base field k of characteristic not 2 or 3, following a suggestion of Serre [Se], has recently been established by Rost [Ro1]. In fact, Serre (loc. cit.) has raised the question as to whether this new invariant, called the invariant mod 3, together with the ordinary trace form already employed in the work of Springer [Sp], classifies Albert algebras over k.

Projective Planes, Graphs, and Simple Algebras

Let X be a projective plane of order n. A flag x 0 ∈ l 0 in X determines a graph F which is used to construct a central simple Jordan algebra A of dimension n 3 − 1 over a field k of characteristic 2 with x 2 = 0 for all x ∈ A. When n is even, the group of all algebra automorphisms of A is isomorphic to the symplectic group Sp( n 3, k). We also imbed every finite connected graph G whose adjacency matrix is non-singular mod 2 in a strongly regular graph G*, and we construct a central simple Lie algebra over Z /2 Z from G*; in particular, F gives a central simple Lie algebra.

On maximal subalgebras of central simple Malcev algebras

Let F be a field of characteristic different from two and A a central sim- ple non-Lie Malcev algebra over F. Our purpose is to determine the maximal subalgebras of A (with respect to inclusion). Maximal subalgebras of simple Lie algebras over an algebraically closed field have been studied by Dynkin, [3, 41, and maximal subalgebras of central simple associative, alternative and special Jordan algebras have been studied by Racine [ 121. Racine’s results on octonion algebras will be relevant in our investigation. 1.

A Cayley-Dickson Process for a Class of Structurable Algebras

In this paper, we study the class of all simple structurable algebras with the property that the space of skew-hermitian elements has dimension $1$. These algebras with involution have arisen in the study of Lie algebra constructions. The reduced algebras are isotopic to $2 \times 2$ matrix algebras. We study a Cayley-Dickson process for rationally constructing some algebras in the class including division algebras and nonreduced nondivision algebras. An important special case of the process endows the direct sum of two copies of a $28$-dimensional degree $4$ central simple Jordan algebra $\mathcal {B}$ with the structure of an algebra with involution. In preparatory work, we obtain a procedure for giving the space ${\mathcal {B}_0}$ of trace zero elements of any such Jordan algebra $\mathcal {B}$ the structure of a $27$-dimensional exceptional Jordan algebra.

Duality of observables and generators in classical and quantum mechnics

In classical and in quantum mechanics physical variables play a dual role as observables and as generators of infinitesimal transformations in the invariance groups. We show that if the Lie algebra of generators is central simple, the observable-generator duality restricts the structure of the algebra of observables to two cases: a commutative, associative algebra as in classical mechanics, or a central simple special Jordan algebra as in quantum mechanics.

The Freudenthal-Springer-Tits constructions revisited

In [3] some constructions of exceptional Jordan algebras due to H. Freudenthal, T. A. Springer, and J. Tits were carried over to quadratic Jordan algebras (as in [4]) of arbitrary characteristic. The question was left open whether the Tits Constructions yielded all exceptional finite-dimensional central simple algebras in characteristic 2 (it was known that they do for characteristics #2). In this paper we settle this question in the affirmative. This result completes the structure theory for finite-dimensional quadratic Jordan algebras. The Tits Constructions as given in [3] involve the construction of a norm form. To prove that all the exceptional algebras arise from these constructions we need to show that all such algebras have a suitable norm form. This necessitates a slight detour in ??1 and 2 to verify that Jordan algebras in characteristic 2 have generic norms with the same properties as in the other characteristics. In ?3 we define the centroid for quadratic Jordan algebras and the corresponding notion of central simple algebras. In the next section we establish certain conditions under which a central simple Jordan algebra remains simple upon extension of the base field. In ?5 we apply these results to show that every exceptional finite-dimensional central simple algebra is a form of the 27-dimensional exceptional algebra ?((i3) (i.e. becomes ( upon suitable extension of the base field). In the final section we show that the Tits Constructions yield precisely all the exceptional finite-dimensional central simple Jordan algebras. Our proofs will be valid for all characteristics.

Involutionen von Jordan-Algebren

In this paper we will give some algebraic results on certain Lie algebras defined by involutions of Jordan algebras. Most of the results will be used elsewhere for applications in analysis. Let A be the (-1)-eigenspace of an involution J≠Id of a central simple Jordan algebra A of degree at least 3, let D be the Lie algebra of all inner derivations of A leaving A_ invariant, and let h be the Lie algebra D+L(A_), where L denotes the regular representation of A. In the case where the 1-eigenspace A+ of J is central simple too, we will show A+=A_A_ and prove that h is semi-simple and irreducible on A. If A+ is not central simple, then A_A_ has codimension 1 in A+ and h is semi-simple, but irreducible on A_A_+A_ only if the characteristic of the groundfield does not divide the degree of A. At characteristic O we will view D as an extension of the derivationalgebra of A+ and determine the structure of the kernel of this extension.

Automorphism Groups of Jordan Algebras

In his development of a structure theory for Jordan algebras of characteristic two, E.C. Paige [1] introduces an important class of central simple Jordan algebras S[2n], It is the purpose of this paper to completely determine the automorphism groups of the algebras S[2n]. The automorphisms will be represented as matrices operating on a natural basis for the underlying vector space of the algebra. Using this characterization, generators and relations will be obtained for each of the automorphism groups. In this way, we will produce an infinite family of finite 2-groups.

Finite power-associative division rings

The classical Wedderburn theorem [5, p. 37] states that any finite associative division ring is a (commutative) field. A. A. Albert generalized this to finite strictly power-associative division rings of characteristic '-2. His proof used the classification of central simple Jordan algebras and proceeded by case-checking (types A, B, C, D in [1, p. 301] and type ? in [2, p. 11]). The purpose of this paper is to give a uniform proof of his results. Throughout the paper all algebras will be nonassociative algebras over a field 4 of characteristic 3 2; since simple rings (in particular, division rings) are simple algebras over their centroids there is no loss in generality in restricting ourselves to algebras. An algebra is a division algebra if left and right multiplications by a nonzero element are bijections; for finite-dimensional algebras this is equivalent to the nonexistence of proper zero divisors. Following N. Jacobson, we define a Jordan division algebra to be a commutative Jordan algebra with identity element such that every nonzero element x is regular with Jordan inverse y: xy = 1, x2y = x. For special algebras the inverse is just the usual inverse in the associative sense. An algebraic Jordan algebra is a Jordan division algebra if and only if each nonzero x generates a subfield 4 [x], the inverse being a polynomial in x [7, p. 1157]. (Note that this condition is weaker than being a division algebra-if e is an associative quaternion division algebra then D+ is a Jordan division algebra with zero divisors). The following lemma is due to Albert [1, p. 299].

On a class of quadratic algebras

We shall denote this class by e. The commutative algebras in e are the well-known central simple Jordan algebras of degree two. In [3] an example is given (over F containing v-I of a 7-dimensional algebra in e which is not commutative. We shall denote by C' the class of all algebras in C which are not commutative. In this note we show that, in the presence of the other conditions on the algebras in C, identity (1) may be replaced by

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

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.

A theorem on the derivations of Jordan algebras

The restriction on the dimensionality of the simple components arises from the fact that the (3-dimensional) central simple Jordan algebra of all 2 X 2 symmetric matrices has for its derivation algebra the abelian Lie algebra of dimension 1. However, most simple Jordan algebras over F have simple derivation algebras, and all except those of dimension 3 over their centers have derivation algebras which are semisimple or {0 }, as may be seen from the listing by N. Jacobson [3, ?4] of these derivation algebras.2 The part of the theorem then follows from the direct sum relationship. To demonstrate the converse it is sufficient to show that, if Z is semisimple or {0}, then 2( is semisimple. For then, if any simple component of 2t had dimension 3 over its center, it would have an abelian derivation algebra not equal to {0 } [3, ?4], which would give rise to a nonzero abelian ideal in Z, a contradiction. To show that 2t is semisimple whenever Z is semisimple or {0}, we use the so-called Wedderburn Principal Theorem for Jordan algebras, proved recently by A. J. Penico [5]. Also Lemma 1 below is taken from the proof of that theorem [5, ?2]. Revisions have been made in the proof of our theorem in accordance with helpful suggestions of Professor Jacobson.