Quadratic Jordan Research Articles

Generically algebraic Jordan algebras over commutative rings

The theory of the generic minimum polynomial, norm and trace is developed for quadratic Jordan algebras which are finitely generated and projective modules over an arbitrary commutative base ring, using scheme-theoretic methods. We recover, with new proofs, most of the classical theory over fields, and also obtain a number of results which are new even in the classical setting.

The Scalar Center for Quadratic Jordan Algebras#

We propose that an element of a Jordan algebra J should be considered “central” if it is a scalar multiple of 1 in some tight unital extension of J. For Jordan algebras with no trivial ideals, this yields an acceptable center. Here it is important that an algebra with no extreme elements has a unique tight unital hull. It has been noted by Fernández-López, García-Rus and Montaner that a necessary condition for centrality of an element c is that the operators U c , V c must be in the centroid. When the algebra is nondegenerate, we show this condition is actually sufficient.

Grassmann speciality of Jordan supersystems

The Grassmann envelope Γ(S)=S 0 ̄ ⊗ Γ 0 ̄ +S 1 ̄ ⊗ Γ 1 ̄ is a useful tool for deciding varietal questions about supersystems S=S 0 ̄ +S 1 ̄ . It is not useful in deciding simplicity, since the envelope is always fraught with trivial ideals coming from trivial superscalars. At first glance it would not seem useful in deciding superspeciality either, but we will show that it is a more sensitive tool than it seems. We say a Jordan supersystem J s (algebra, triple, or pair) is Grassmann special if its Grassmann envelope is special as an ordinary Jordan system over Γ 0 ̄ . Certainly superspeciality J s ⊂ A s + over Φ implies Grassmann speciality Γ( J s )⊂Γ( A s ) + over Γ 0 ̄ , but it is not obvious how Grassmann speciality influences superspeciality of J s . Nevertheless, we show how to transform obstacles to speciality of J s over Φ (anti-superspecial elements) into obstacles to speciality of Γ( J s ) over Γ 0 ̄ (anti-special elements) using Grassmann boosters, so that non-superspeciality J s over Φ implies non-speciality of Γ( J s ) over Γ 0 ̄ . Thus Grassmann speciality is the same as superspeciality: a supersystem J is a superspecial Jordan supersystem over Φ iff Γ( J s ) is a special Jordan system over Γ 0 ̄ . (It is not clear this is the same as speciality of Γ( J s ) over Φ, since in general speciality of quadratic Jordan systems depends on the scalars: we give an example of a Jordan Ω -algebra which is not special, but is special over a scalar subring Φ.) As a corollary, any Jordan superalgebra with zero linear extreme radical which is viably evenly 4-interconnected is superspecial; thus exceptional superalgebras must be built over even parts of degree ⩽3.

Quadratic Jordan superpairs covered by grids

In this paper we are proposing a theory of Jordan superpairs defined over (super)commutative superrings. Our framework has two novelties: we allow scalars of even and odd parity and we do not assume that 1 2 lies in our base superring. To demonstrate that it is possible to work in this generality we classify Jordan superpairs covered by a grid.

Computational approach to the simplicity of f4( Os,−) in the characteristic two case

We present in this paper a computational approach to the study of the simplicity of the derivation Lie algebra of the quadratic Jordan algebra H 3( O s ,−), denoted by f 4( O s ,−), when the characteristic of the base field is two. We will show not only a collection of routines designed to find identities and construct principal ideals but also a philosophy of how to proceed studying the simplicity of a Lie algebra. We have first implemented the quadratic Jordan structure of H 3( O s ,−) into the computer system Mathematica (Computing the derivation Lie algebra of the quadratic Jordon Algebra H 3( O s ,−) at any characteristic, preprint, 2001) and then determined the generic expression of an element of the Lie algebra f 4(O s,−)= Der(H 3(O s,−)) (see (41)). Once the structure of f 4(O s,−) is completely described, it is time to analyze the simplicity by using the strategy mentioned. If the characteristic of the base field is not two, the Lie algebra is simple, but if the characteristic is two, the Lie algebra is not simple and there exists only one proper nonzero ideal I which is 26 dimensional and simple as a Lie algebra. In order to prove this last affirmation, we have used again the set of routines to show the simplicity of the ideal and that it is isomorphic to f 4/I , which is also a simple Lie algebra. This isomorphism is constructed from a computed Cartan decomposition of both Lie algebras.

SPECIALITY OF INTERCONNECTED QUADRATIC JORDAN ALGEBRAS

ABSTRACT A non-unital quadratic Jordan algebra J with viable n-intercon nectivity () and zero extreme radical is shown to be special. If we assume that J is a simple (non-unital) quadratic Jordan algebra with at least 3 non-zero, non-supplementary orthogonal idempotents, then we show directly (independent from the McCrimmon-Zelmanov classification of simple algebras [7]) that J already is viably 4-interconnected with zero extreme radical, hence is special.

Goldie Theory for Jordan Algebras

It is shown that Zelmanov's version of Goldie's conditions still characterizes quadratic Jordan algebras having an artinian algebra of quotients which is nondegenerate. At the same time, Jordan versions of the main notions of the associative theory, such as those of the uniform ideal, uniform element, singular ideal, and uniform dimension, are studied. Moreover, it is proved that the nondegenerate unital Jordan algebras of finite capacity are precisely the algebras of quotients of nondegenerate Jordan algebras having the property that an inner ideal is essential if and only if it contains an injective element.

QUADRATIC JORDAN SUPERALGEBRAS

This paper examines Kevin McCrimmon's proposed definition for a quadratic Jordan superalgebra meaningful over an arbitrary ring of scalars. Efim Zel'manov has proposed two conditions that such a definition should satisfy: first, all the simple, finite-dimensional, linear Jordan superalgebras of characteristic 0 in Kac' classification should have simple, quadratic Jordan superalgebra analogs, and second, these should be essentially all the simple, finite-dimensional, quadratic Jordan superalgebras of characteristic 0. Here we verify the first condition. Specifically we study the quadratic superalgebra analogues of full linear superalgebras, Hermitian superalgebras, Kaplansky's superalgebras, Kac's split 10-dimensional superalgebra, and superalgebras of quadratic forms with base point.

The Structure of Quadratic Jordan Systems of Clifford Type

The Structure of Quadratic Jordan Systems of Clifford Type

On the absolute stability of a class of indirect control systems

The absolute stability of a class of indirect control systems was studied by applying the theory of Hermitian quadratic form and Jordan normal form. The algebraic formal criteria for the absolute stability are established, and these results are new and useful.

Jordan centroids

Central simple triples are important for the classification of prime Jordan triples of Clifford type in arbitrary characterstics. For triples and pairs (or even for unital Jordan algebras of characteristic 2), there is no workable notion of center, and the concept of “central simple” system must be understood as “centroid-simple”. The centroid of a Jordan system (algebra, triple, or pair) consists of the “natural” scalars for that system: the largest unital, commutative ring Γ such that the system can be considered as a quadratic Jordan system over Γ. In this paper we will characterize the centroids of the basic simple Jordan algebras, triples, and pairs. (Consideration of the tangled ample outer ideals in Jordan algebras of quadratic forms will be left to a separate paper.) A powerful tool is the Eigenvalue Lemma, that a centroidal transformation on a prime system over φ which has an eigenvalue α in φ must actually be scalar multiplication by α. An important consequence is that a prime system over φ with reduced elements PxJ = φx (or which grows reduced elements under controlled conditions) must already be central, Γ = φ.

Little jordan clofford algebras

Unital quadratic Jordan algebras J(q, I) determined by nondegenerate quadratic forms with basepoints over a field axe called full Jordan Clifford algebras. In characteristic 2 they have ample outer ideals which are also simple; they come in 3 sizes, tiny, small, and large, where the large are full Clifford algebras but the tiny and small algebras are lacking some of their parts. The simple algebras played a role in Zelmanov's solution of the Burnside Problem. In this paper we will analyze these in more detail, determining their centroids and their local algebras; this is important in the classification of prime Jordan triples of Clifford type in arbitrary char-acterstics. In addition we make a careful charcterization of the tiny, small, and large Clifford algebras. We use this to straighten one or two missteps in a proof from the classification of simple algebras. An important role in our characterization is played by commutators, and we describe the Jor­dan commutator products and Bergmann formulas for Clifford algebras in general.

Primitive Jordan Pairs and Triple Systems

In this paper we give a characterization of primitivity of Jordan pairs and triple systems in terms of their local algebras. As a consequence of that local characterization we extend to Jordan pairs and triple systems most of the known results about primitive Jordan algebras. In particular, we describe primitive Jordan pairs and triple systems over an arbitrary ring of scalars in the sense of “The Structure of Primitive Quadratic Jordan Algebras” by J. A. Anquela, T. Cortés, and F. Montaner (1995,J. Algebra172,530–553, 5.1).

The structure of primitive quadratic Jordan algebras

In this paper we give a complete description of primitive quadratic Jordan algebras following the classification of strongly prime quadratic Jordan algebras (K. McCrimmon and E. I. Zel'manov, Adv. Math. 69, No. 2 (1988), 133–222). The proof is based in two results of independent interest: we show that every P. I. primitive Jordan algebra is simple and unital with nonzero socle and prove that associative tight (*-tight) envelopes of special primitive Jordan algebras are also primitive (*-primitive). As a consequence of the latter fact we see that an associative algebra A is one-sided primitive if and only if A + is primitive, and an associative algebra A with involution * is *-primitive if and only if an ample subspace H 0 ( A , *) is primitive.

Primitivity in Finitely Generated Special Quadratic Jordan Algebras

Primitive Jordan algebras were introduced by Zelmanov in the linear case [13] and Hogben and McCrimmon in the quadratic case [4]. A Classification Theorem for quadratic primitive Jordan algebras, in the spirit of the general classification of prime nondegenerate Jordan algebras [9], is proven in [2]. In that result, besides Albert- and Clifford-type algebras, there appear special algebras containing an ideal of the form H 0( A, ∗) for a primitive associative algebra A with involution. To prove this result, a key fact is that, for a prime A, primitivity can be transferred from H 0( A, ∗) to A and back. The first part of this result stems from a more general theorem asserting that any tight associative envelope of a primitive Jordan algebra is primitive [2]. This can be extended to ∗-envelopes making use of the concept of ∗-primitivity: any ∗-tight associative envelope of a primitive Jordan algebra is ∗-primitive. The aim of this paper is to examine the reciprocal of that theorem for finitely generated Jordan algebras. We prove in Section 4 that a finitely generated special Jordan algebra having a ∗-primitive ∗-envelope is primitive. In the case of H( A, ∗) the proof makes uses of the Herstein second construction [7], for a semiprime A with involution ∗, any nonzero ideal of H( A, ∗) contains some H 0( I, ∗) with I a nonzero ∗-ideal of A. Our proof will follow a similar pattern. We prove in Section 2 that the Herstein-McCrimmon construction can be extended to prime finitely generated Jordan algebras with nonzero hermitian part. For that we make use of some basic properties of hermitian ideals as defined in [9]. In Section 3 we treat algebras an anti-hermitian type (called here simply anti-hermitian algebras) separately and we collect the previous results in Section 4 to prove our main theorem.

Annihilators of elements of the socle of a jordan algebra

The double annihilator of any element in the socle of a nondegenerate quadratic Jordan algebra coincides with the principal inner ideal generated by this element. Since the socle satisfies dec on principal inner ideals, it also satisfies ace on annihilators of principal inner ideals.

Quadratic Jordan systems of hermitian type

In this paper, we extend Zel'manov's classification of linear Jordan systems (triple systems and pairs) of hermitian type to quadratic Jordan systems over an arbitrary ring of scalars. Using associative triple systems (of the first kind) for what we believe to be a more natural tool to describe special Jordan triples, we show that an i-special prime quadratic Jordan triple system (the ideals of which remain semiprime) with a nonzero hermitian part lies between an ample subspace of hermitian elements in a ∗-prime associative triple system and those in its Martindale system of symmetric quotients. Modulo some extra definitions, the structure of strongly prime Jordan pairs of hermitian type follows almost immediately from the Jordan triple result.

Zel'manov polynomials in quadratic Jordan triple systems

Zel'manov polynomials are elements of a free special Jordan system which are both hermitian (their values look like hermitian elements) and Clifford (they do not vanish on systems T containing H 3 = H( M 3( Φ), t)). These polynomials decide the classification of strongly prime special Jordan systems: T is either Clifford or hermitian according as some Zel'manov polynomial does or does not vanish on T. The existence of such polynomials has been established by McCrimmon and Zel'manov for quadratic Jordan algebras, and by Zel'manov for linear Jordan triple systems (and pairs). In this paper, we carry out the construction for quadratic Jordan triple systems (and pairs).

Order Relation in Quadratic Jordan Rings and a Structure Theorem

Order Relation in Quadratic Jordan Rings and a Structure Theorem

Coordinatization of triangulated Jordan systems

We obtain partial coordinatizations of any quadratic Jordan triple system containing a triangle (e,, U, e,), i.e., a “split” Jo = ke, @ ku 0 ke, z Hz(k), such that e=e, +ez is invertible and u is faithful (in the sense that L(x, , e, )u = 0 + x, = 0; this is automatic if J is nondegenerate or t E k). We identify a hermitian matrix subsystem Jh = J,(e,)@ Du@ J2(e2) E H,(D, D,, 7t, ~ ) and a Clifford subsystem J,, = K, @NO K, E J(q, S, C,). We show that a simple J must be all hermitian or all Clifford (coincides with J,, or J,), yielding a shorter proof Osborn’s Capacity 2 Theorem for Jordan algebras. We also can coordinatize most unital bimodules for hermitian matrix systems. The impetus for this study came from two sources. In presenting a selfcontained treatment of Jordan structure theory, N. Jacobson [4] sought a shorter proof of the Osborn Capacity 2 Theorem; the proof is not short for linear Jordan algebras [2] and is quite involved for quadratic Jordan algebras [3]. In coordinatizing Jordan triple systems, the second author needed a version of the Capacity 2 Theorem making no semisimplicity restriction on the ambient system and without assuming that the tripotents e, are division tripotents [lo]. By avoiding semisimplicity assumptions, such a formulation also coordinatizes unital bimodules. In the triangulated case one cannot expect as precise a coordinatization as for rank n > 3: In the latter case if J is a Jordan algebra then Jz