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A 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 We also obtain explicit formulas for the root groups, the

Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or \"form\" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a \"residually pseudo-split\" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms.This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings.

Let G be an absolutely almost simple algebraic group over a field K. The genus genK​(G) of G is the set of K-isomorphism classes of K-forms G′ of G that have the same K-isomorphism classes of maximal K-tori as G. We construct an example of outer forms of type A2​ with infinite genus.The submission date of this paper had been incorrectly displayed on the web page between 2 July 2024 and 5 June 2025. For the details, see the erratum.

We describe the construction of a specific class of disconnected locally compact near-fields. They are so-called Dickson near-fields and derived from p -adic division algebras by means of a special kind of homomorphisms or antihomomorphisms from the multiplicative group into the group of inner automorphisms of the division algebra. So let F be a local field and D be a finite-dimensional central division algebra over F . We presuppose that D / F is tamely ramified. In the first part of this paper we determine all finite subgroups of D ∗ / F ∗ . Based on that, we then determine all homomorphic and antihomomorphic couplings D ∗ → Inn ( D ) = D ∗ / F ∗ with finite image. With each of these couplings a locally compact near-field can be constructed from D . Apart from isomorphism, there is only a finite number of them. Compared to a previous publication, we omit the assumption that the image of the couplings is an Abelian group.

We give a classification of the group gradings on M p ( K ) , here p is a prime number and K is a field that contains a primitive p -th root of unity if char K ≠ p . The gradings are isomorphic to an elementary grading, to a division grading with support isomorphic to Z p or to a division grading with support isomorphic to Z p × Z p .

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\\lambda \\in D$ for which $f^{\\circ n}(\\lambda )=\\lambda $ for any $n \\in \\mathbb {N}$ , where $f^{\\circ n}(x)$ is defined recursively by $f^{\\circ n}(x)=f(f^{\\circ (n-1)}(x))$ and $f^{\\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\\lambda $ is a fixed point of $f(x)$ if and only if $f(\\lambda )=\\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \\geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

In this note, we present a chain lemma for biquaternion algebras over fields of characteristic 2 in the style of the equivalent chain lemma by Sivatski in characteristic not 2, and conclude a bound on the symbol length of classes in 2 n B r ( F ) whose symbol length in 2 n + 1 B r ( F ) is at most 4.

We establish results on additive maps preserving inverses in the context of alternative division algebras. A related discussion on semi-automorphisms and Jordan homomorphisms is included. Furthermore, we pose some open questions about the algebraic structure of the set of Jordan homomorphisms of alternative division algebras of characteristic 2.

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley–Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the “imaginary tower”), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call “half-quaternions” and “half-octonions”. The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary “half-octonions” is unitless and totally associative.

Nonassociative differential extensions are generalizations of associative differential extensions, either of a purely inseparable field extension K of exponent one of a field F , F of characteristic p , or of a central division algebra over a purely inseparable field extension of F . Associative differential extensions are well known central simple algebras first defined by Amitsur and Jacobson. We explicitly compute the automorphisms of nonassociative differential extensions. These are canonically obtained by restricting automorphisms of the differential polynomial ring used in the construction of the algebra. In particular, we obtain descriptions for the automorphisms of associative differential extensions of D and K , which are known to be inner.