Noncommutative Jordan Research Articles

Quadratic Jordan algebras and cubing operations

In this paper we show how the Jordan structure can be derived from the squaring and cubing operations in a quadratic Jordan algebra, and give an alternate axiomatization of unital quadratic Jordan algebras in terms of operator identities involving only a single variable. Using this we define nonunital quadratic Jordan algebras and show they can be imbedded in unital algebras. We show that a noncommutative Jordan algebra A \mathfrak {A} (over an arbitrary ring of scalars) determines a quadratic Jordan algebra A + {\mathfrak {A}^ + } .

Nodal algebras of dimension 𝑝³

An algebra A is in K if and only if there exist p, n, and P such that A+ = Bn where multiplication in A is defined by 1 » df dg 2 ,-,/_i dXi OX/ in terms of the (dot) product in Bn, and the [xt, Xj]=XiX/—XjXi are arbitrary except for the condition that at least one of these commutators be nonsingular. Every simple nodal noncommutative Jordan algebra is in K, but not all the algebras in K are simple [3]. Properties of K have been studied in [2], [3], [4], and in several other papers, but the ideal structure of the nonsimple algebras of K is unknown. In this paper we give a complete list of the ideals of a Lie-admissible algebra in K with three generators. Theorem. // A EK is Lie-admissible and of dimension ps, then there exist generators x, y, z such that A = Fl-\-F[x, y, z] (vector space direct sum) with [y, x] = 1 + c-xp-l-yp~1, c E A, (1) [z, x] = y'T~1-m(z), m(z) E P[l, z], and [z, y] = x*-l-n(z), n(z) E F[l, z].

Noncommutative Jordan Algebras with Commutators Satisfying an Alternativity Condition

The theorems of this paper show that the main results in the structure and representation theory of Jordan algebras and of alternative algebras are valid for a larger class of algebras defined by simple identities which obviously hold in the Jordan and alternative cass. A new unification of the Jordan and associative theories is also achieved.

A Norm on Separable Noncommutative Jordan Algebras of Degree 2

A Norm on Separable Noncommutative Jordan Algebras of Degree 2

A norm on separable noncommutative Jordan algebras of degree $2$

A norm on separable noncommutative Jordan algebras of degree $2$

A class of flexible nilstable algebras

have been studied by Kosier [2 ] and by the author [3]. When such an algebra is strictly power-associative over a field of characteristic prime to six then, modulo its radical, it has an identity and is a direct sum of simple algebras. The simple algebras of degree no less than three are Jordan or quasi-associative. Those of degree two and nilstable were shown in [3] to be noncommutative Jordan algebras that are J-simple and consequently in possession of a multiplication table like that described by Kokoris [1]. In this paper we shall observe that when the ground field is ordered such an algebra is in fact Jordan and point out noncommutative examples of such algebras when the ground field is not ordered.

On a Class of Nodal Noncommutative Jordan Algebras

On a Class of Nodal Noncommutative Jordan Algebras

On a class of nodal noncommutative Jordan algebras

where f-g is the product of f=f(x1, . . ., xn) and g=g(x1, . . ., xn) in Bn and the = 1[xi, xj] = (xixj -xjxi) are arbitrary except for the proviso that at least one of them is nonsingular. That is, there must exist a cij = aij1 + wij with aij =0. This implies that n ? 2. The class K was constructed by L. Kokoris who proved [2], [3] that every simple nodal noncommutative Jordan algebra is in K. He also proved that not all the algebras of the class K are simple. Two papers have appeared which were concerned with studying derivations of some of the algebras in K. The first of these by R. Schafer [7] described derivation algebras for the cases cij in F and n =2 and demonstrated relationships between certain ideals of these algebras and types of simple Lie algebras of characteristic p. He made use of two properties which were later generalized by R. Oehmke, namely that the algebras in K for the cases cij in F and n =2 are Lie-admissible and that, for n = 2, the generators xl and x2 could be chosen so that c12 = 1 + axp -1*x2-1 for a in F. In his paper [4], Oehmke determined the derivation algebras of all simple Lieadmissible algebras A of K. He proved that generators could be chosen for these A which satisfy a useful Lie-multiplication table, which is a generalization of

A proof of Schafer's conjecture for infinite-dimensional forms admitting composition

is necessarily a separable finite-dimensional alternative algebra, and that Q is a product of irreducible factors of the generic norm. In [Z2] he showed such an algebra is necessarily alternative, is separable when finite-dimensional, and in this case (using results of Professor N. Jacobson [4]) that Q is a product of irreducible factors of the generic norm, so the problem reduced to proving finite-dimensionality. This was a classical result if Q was of degree 2; such an algebra was a composition algebra, and these could be explicitly constructed [2] and shown to have dimension 1,2,4, or 8. There is little hope of generalizing this method to Q of arbitrary degree, since it would involve an algorithm for constructing all separable alternative algebras. Schafer’s approach was more akin to Professor I. Kaplansky’s original proof [6] of the degree 2 case. Kaplansky linearized the basic identity to show first that the algebra was alternative, next that it was algebraic, and finally that it was semisimple. At this juncture results on alternative rings were invoked to conclude that the algebra was finite-dimensional. In this paper we will apply a similar procedure to normed algebras [S]. Replacing linearization techniques by differential operators, we will show first that such an algebra is a noncommutative Jordan algebra (Lemma l), next that it is algebraic (Lemma 2), and finally (Theorem 1) that is a direct sum of a finite number of simple ideals which are either

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 .

Structure and Representations of Noncommutative Jordan Algebras

Structure and Representations of Noncommutative Jordan Algebras

Structure and representations of noncommutative Jordan algebras

Structure and representations of noncommutative Jordan algebras

Norms and noncommutative Jordan algebras

Norms and noncommutative Jordan algebras

Nodal noncommutative Jordan algebras

1. A finite-dimensional power-associative algebra ' is said to be nodal [6] if every element of V can be written as a I + z where ai E c 1 is the unity element of W and z is nilpotent and if the set of all nilpotent elements is not a subalgebra of W. In [3; 4], Kokoris has shown that every simple nodal noncommutative Jordan algebra of characteristic p 5 2 has the form 1 = a1 + 9 with W = a[xl, **, x] for some n where the generators are all nilpotent of index p and the multiplication is associative. Iff and g are two elements of W then the multiplication table of 91 is given by

On Anti-Commutative Algebras with an Invariant Form

In this paper we consider anti-commutative algebras with an invariant form, that is, an algebra A over a field F such thatand A possesses a symmetric bilinear form f(x, y) such thatLie and Malcev algebras (2, 3) are examples of such algebras and we shall consider generalizations of these algebras obtained by introducing commutation, x o y = xy — yx, as a new multiplicative operation in the non-commutative Jordan algebras of (1).

On a class of nonflexible algebras

for fixed ai CF. II. There is an algebra B over F such that B satisfies (1), B has an identity element, and B is nonflexible; that is, there are elements x and y in B such that x(yx) # (xy)x. These conditions are similar to those used by Albert to define almost left alternative algebras [5 ]. Albert's paper led to the study of algebras of (y, 8) type by Kleinfeld and Kokoris [9; 1 1; 12 ]. Kokoris has shown that any simple finite dimensional algebra of characteristic prime to 30 of ('y, 8) type is either alternative or has an identity element which is an absolutely primitive idempotent(2). Our alteration of Albert's conditions yields a new class of simple power-associative algebras. We note that property II seems more natural in light of Oehmke's results [13] and the remark that most of the well-known nonassociative algebras (Jordan, noncommutative Jordan, Lie, alternative, associative) satisfy the flexible identity x(yx) = (xy)x. In ?2 it is shown that if F is algebraically closed then any algebra A over F belonging to W is quasi-equivalent in F to an algebra A (yA) where A (y) satisfies one of the following identities: (i) (xy)z-x(yz) = (zy)x-z(yx), (ii) x(xy) +(yx)x= 2(xy)x, (iii) x(xy) + (yx)x = (xy)x+x(yx), (iv) x(yz+zy) +(yz+zy)x=(xy+yx)z+z(xy+yx). The remainder of the paper is devoted to the study of algebras which satisfy some one of these four identities. We find that any power-associative

Some Lie Admissible Algebras

Several studies have been made to obtain larger classes of non-associative algebras from classes of algebras with a known structure. Thus, we have right alternative algebras (2)* and non-commutative Jordan algebras (6), (7), (8), and (9). These algebras are defined by a subset of the set of identities of the algebras from which they derive their names. Also, Albert (1), among others has studied Jordan admissible algebras. This paper is concerned with algebras which are related to Lie algebras in that they satisfy some of the identities of a Lie algebra and are Lie admissible. Theorem 2 answers a question raised by Albert in (1).

On flexible power-associative algebras of degree two

Let 2t be a simple, flexible, power-associative algebra of degree two over an algebraically closed field 0 of characteristic not equal to 2, 3 or 5. Then 2f has a unity element [5, Theorem 3.5] 1 =u+v where u and v are absolutely primitive orthogonal idempotents of Wf. The algebra W can be decomposed as a vector space direct sum t = 2ff(2) +Sfu(1)+Sfu(O) where x is in 2Iu(X) if and only if ux+xu=XX.2 If Sf(X) f,(1) + fu(1) f,(X) C Wf(1) for X = 2, 0, u is called a stable idempotent and 2f is said to be u-stable. If 2L is u-stable for every idempotent u in 2f then 2f is called a stable algebra. In ?1 it is shown that if 2f is a u-stable algebra then there is an element w in Sf(1) such that w2 = 1. The existence of this element is used in ?2 to develop some of the multiplicative properties of Wf. Finally in ?3 it is proved that every simple, flexible, stable, powerassociative algebra of degree two over an algebraically closed field H of characteristic 5&2, 3, 5 is a noncommutative Jordan algebra.

Nodal Noncommutative Jordan Algebras and Simple Lie Algebras of Characteristic p

Nodal Noncommutative Jordan Algebras and Simple Lie Algebras of Characteristic p

Nodal Non-Commutative Jordan Algebras

A finite dimensional power-associative algebra 𝒰 with a unity element 1 over a field J is called a nodal algebra by Schafer (7) if every element of 𝒰 has the form α1 + z where α is in J, z is nilpotent, and if 𝒰 does not have the form 𝒰 = ℐ1 + n with n a nil subalgebra of 𝒰. An algebra SI is called a non-commutative Jordan algebra if 𝒰 is flexible and 𝒰+ is a Jordan algebra. Some examples of nodal non-commutative Jordan algebras were given in (5) and it was proved in (6) that if 𝒰 is a simple nodal noncommutative Jordan algebra of characteristic not 2, then 𝒰+ is associative. In this paper we describe all simple nodal non-commutative Jordan algebras of characteristic not 2.