Jordan Division Algebras Research Articles

Special Moufang sets coming from quadratic Jordan division algebras

The theory of Moufang sets essentially deals with groups having a split BN-pair of rank one. Every quadratic Jordan division algebra gives rise to a Moufang set such that its root groups are abelian and a certain condition called special is satisfied. It is a major open question if also the converse is true, i.e. if every special Moufang set with abelian root groups comes from a quadratic Jordan division algebra. De Medts and Segev [Amer. Math. Soc. 360 (2008), pp. 5831–5852] proved in Theorem 5.11 that this is the case for special Moufang set satisfying two conditions. In this paper we prove that these conditions are in fact equivalent and hence either of them suffices. Even more, we can replace them by weaker conditions.

Cubic Action of a Rank one Group

We consider a rank one group G = ⟨ A , B ⟩ G = \langle A,B \rangle acting cubically on a module V V , this means [ V , A , A , A ] = 0 [V,A,A,A] =0 but [ V , G , G , G ] ≠ 0 [V,G,G,G] \ne 0 . We have to distinguish whether the group A 0 := C A ( [ V , A ] ) ∩ C A ( V / C V ( A ) ) A_0 :=C_A([V,A]) \cap C_A(V/C_V(A)) is trivial or not. We show that if A 0 A_0 is trivial, G G is a rank one group associated to a quadratic Jordan division algebra. If A 0 A_0 is not trivial (which is always the case if A A is not abelian), then A 0 A_0 defines a subgroup G 0 G_0 of G G acting quadratically on V V . We will call G 0 G_0 the quadratic kernel of G G . By a result of Timmesfeld we have G 0 ≅ S L 2 ( J , R ) G_0 \cong \mathrm {SL}_2(J,R) for a ring R R and a special quadratic Jordan division algebra J ⊆ R J \subseteq R . We show that J J is either a Jordan algebra contained in a commutative field or a Hermitian Jordan algebra. In the second case G G is the special unitary group of a pseudo-quadratic form π \pi of Witt index 1 1 , in the first case G G is the rank one group for a Freudenthal triple system. These results imply that if ( V , G ) (V,G) is a quadratic pair such that no two distinct root groups commute and c h a r V ≠ 2 , 3 \mathrm {char} V\ne 2,3 , then G G is a unitary group or an exceptional algebraic group.

Moufang Sets and Structurable Division Algebras

A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. We extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, we show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k k of characteristic different from 2 2 and 3 3 arises from a structurable division algebra. We also obtain explicit formulas for the root groups, the τ \tau -map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.

Linearly Minimal Jordan Algebras of Characteristic Other Than 2

It is proved that every nontrivial linearly minimal Jordan algebra of characteristic other than 2 is a division algebra. Then Zel’manov’s classification of Jordan division algebras is applied to show that such an algebra is a field.

On the identities defining a quadratic Jordan division algebra

Quadratic Jordan algebras are defined by identities that have to hold strictly, i.e that continue to hold in every scalar extension. In this paper we show that strictness is not required for quadratic Jordan division algebras.

Algebraic structures related to universal covers of special rank one groups

In this note we give a presentation of special rank one groups with abelian unipotent subgroups (abbreviated AUS), depending on some algebraic structure K, and then show that each special rank one group with AUS is a center-factor-group of a so presented group. As shown in [T. De Medts, R. Weiss, Moufang sets and Jordan division algebras, Math. Ann. 335 (2006) 415–433] one example of such an algebraic structure is unital quadratic Jordan Division Algebras. In subsequent work my student Sebastian Weiß studied these structures, called R-structures, systematically and showed, among other things, that the problem of classifying such special rank one groups with AUS is equivalent to the classification of these R-structures up to Isotopy.

Proper Moufang sets with abelian root groups are special

Moufang sets are split BN -pairs of rank one, or the Moufang buildings of rank one. As such they have been studied extensively being the basic ‘building blocks’ of all split BN -pairs. A Moufang set is proper if it is not sharply 2-transitive. We prove that a proper Moufang set whose root groups are abelian is special. This resolves an important conjecture in the area of Moufang sets. It enables us to apply the theory of quadratic Jordan division algebras to such Moufang sets.

Quadratic rank-one groups and quadratic Jordan division algebras

A rank-one group X is a group generated by two distinct nilpotent subgroups A and B such that, for each a ∈ A#, there exists a b ∈ B# satisfyingAb =Ba and vice versa. It has been shown that the notions of a rank-one group and of a group with a splitBN-pair of rank 1 are equivalent. Hence all algebraic groups of relative rank 1 and classical groups of Witt index 1 are rank-one groups. The rank-one group X = 〈 A,B 〉 is said to be quadratic if there exists a ℤX-module V satisfying [V,X, X] ≠ 0 = [V,A, A]. In the main result of this paper we classify the quadratic rank-one groups, that is, we show that there is a one-to-one correspondence of such groups with special quadratic Jordan division algebras.

Cyclic compositions and trisotopies

Cyclic compositions in the sense of Springer [T.A. Springer, Oktaven, Jordan-Algebren und Ausnahmegruppen, Universität Göttingen, 1963] and Knus, Merkurjev, Rost and Tignol [M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, Amer. Math. Soc. Colloq. Publ., vol. 44, Amer. Math. Soc., Providence, RI, 1998. With a preface in French by J. Tits] are investigated by means of cyclic trisotopies, a concept originally due to Albert [A.A. Albert, A construction of exceptional Jordan division algebras, Ann. of Math. (2) 67 (1958) 1–28]. Using the quadrupling of composition algebras, we enumerate cyclic trisotopies and compositions in a rational manner, i.e., without extending the base field. We relate cyclic trisotopies explicitly to simple associative algebras of degree 3 with involution and to the Tits process of cubic Jordan algebras.

Moufang sets and jordan division algebras

We give a simple criterion which determines when a permutation group U and one additional permutation give rise to a Moufang set. We apply this criterion to show that every Jordan division algebra gives rise in a very natural way to a Moufang set, to provide sufficient conditions for a Moufang set to arise from a Jordan division algebra and to give a characterization of the projective Moufang sets over a commutative field of characteristic different from 2.

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.

Albert division algebras in characteristic three contain cyclic cubic subfields

It is shown that a finite-dimensional absolutely simple nonsingular Jordan division algebra of degree 3 over a field containing the third roots of unity admits a cyclic cubic subfield.

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.

Cubic Subfields of Exceptional Simple Jordan Algebras

Let $E/k$ be a cubic field extension and $J$ a simple exceptional Jordan algebra of degree 3 over $k$. Then $E$ is a reducing field of $J$ if and only if $E$ is isomorphic to a (maximal) subfield of some isotope of $J$. If $k$ has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over $k$ contains a cyclic cubic subfield.

Generic reducing fields of Jordan pairs

Generic reducing fields of Jordan pairs, generalizing at the same time generic splitting fields of associative algebras and generic zero fields of quadratic forms, are intrinsically defined and constructed. The most elementary properties are derived, and the relationship with other generic constructions, particularly those linked to Brauer-Severi varieties, are investigated. As an application it is shown that there exist nonisomorphic exceptional Jordan division algebras having the same splitting fields.

Cubic subfields of exceptional simple Jordan algebras

Let E / k E/k be a cubic field extension and J J a simple exceptional Jordan algebra of degree 3 over k k . Then E E is a reducing field of J J if and only if E E is isomorphic to a (maximal) subfield of some isotope of J J . If k k has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over k k contains a cyclic cubic subfield.

Central polynomials for special Jordan rings without nilpotent elements, and an analog of Kaplansky's theorem

In [7], a theory of nonassociative algebras with central polynomial was given, with several applications to Jordan algebras. The main theorem of this paper is that every special Jordan PI-ring J without nilpotent elements satisfies a class of central polynomials (related to the Formanek-Amitsur class of central polynomials for associative rings). This yields a “generic” Hamilton-Cayley equation for J. I f J is a torsion-free module over its center, then we can apply a result of Shirshov [9] (cf. [IS]) to [8], t o see that the “ring of central quotients” of J is simple and is either the Jordan algebra of a symmetric bilinear form or is finite dimensional (over its center), with a bound on the dimension computed in terms of the PI degree. In particular, every special Jordan PI-division ring is finite dimensional. This generalizes part of a result of Smith [ll], that every special algebraic simple Jordan PI-ring is finite dimensional. After this paper was distributed in preprint form, Bokut [17] announced the determination by Zelmanov of all Jordan division algebras of characteristic f2, thereby completing the Jacobson-Osborn structure theory of Jordan algebras with $ satisfying the descending chain condition on inner ideals; a generalization of Theorem 7 would follow immediately from Zelmanov’s results. To prove the main theorem we obtain a criterion for an element a of a special Jordan algebra without nilpotent elements to be central, namely, that [a, a, x] = 0 for all elements x. ([a, b, c] denotes the associator (ab)c a(&).) This is proved by verifying a fact about generalized identities of associative rings with involution. Other notational conventions are: [a, b] denotes ub bu, Z(R) denotes the center (of a ring R) = (a E R 1 [z, R, R] = [R, z, R] = [R, R, z] = [R, z] = 0). Nil ( ) denotes “the” maximal ideal of nilpotent elements, where an element Y is nilpotent if we can take some product of r, under suitable placement of paren-

Prime Jordan P.I. Algebras with nonzero socle and Jordan division algebras

In [ll, p. 4291, Jacobson proposed the problem of establishing a P.I. theory for Jordan rings. In particular he asks whether every simple P.I. Jordan algebra is either finite dimensional or gotten from a nondegenerate quadratic form. Jordan P.I. rings were then investigated by Smith and Rowen [15-17, and their bibliographies] among others. In this paper we investigate the structure of nondegenerate prime Jordan P.I. algebras with nonzero socle (see Section 2 for definitions). Using the classification of prime Jordan rings with nonzero socle established in a preceeding paper [13], we prove (in Section 2)

Exceptional Jordan division algebras over a field with a discrete valuation.

Article Exceptional Jordan division algebras over a field with a discrete valuation. was published on January 1, 1975 in the journal Journal für die reine und angewandte Mathematik (volume 1975, issue 274-275).

Noncommutative Jordan division algebras

The structure theory for noncommutative Jordan algebras with chain conditions leads to the following simple algebras: (I) division algebras, (II) forms of nodal algebras, (III) algebras of generic degree two, (IV) commutative Jordan matrix algebras, (V) quasi-associative algebras. The chain condition is always satisfied in a division algebra, hence does not serve as a finiteness restriction. Consequently, the general structure of noncommutative Jordan division algebras, even commutative Jordan division algebras, is unknown. In this paper we will classify those non-commutative Jordan division algebras which are forms of algebras of types (II)-(V); this includes in particular all the finite-dimensional ones.