Composition Algebras Research Articles

Compact Exceptional Simple Kantor Triple Systems Defined on Tensor Products of Composition Algebras*

In this article we give the classification of compact exceptional simple Kantor triple systems defined on tensor products of composition algebras A = 𝔸 1⊗ 𝔸 2 such that their Kantor algebras ℒ(φ, A) are real forms of exceptional simple Lie algebras.

Composition algebras of the second kind

The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in a countable dimension. The constructed algebras each contains a non-isotropic idempotent e2 = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A, the group Ortaut A of orthogonal automorphisms is specified.

The Magic Square and Symmetric Compositions II

The construction of Freudenthal's Magic Square, which contains the exceptional simple Lie algebras of types F_4,E_6,E_7 and E_8 , in terms of symmetric composition algebras is further developed here. The para-Hurwitz algebras, which form a subclass of the symmetric composition algebras, will be defined, in the split case, in terms of the natural two dimensional module for the simple Lie algebra \mathfrak{sl}_2 . As a consequence, it will be shown how all the Lie algebras in Freudenthal's Magic Square can be constructed, in a unified way, using copies of \mathfrak{sl}_2 and of its natural module.

Levels and Sublevels of Composition Algebras

Abstract Lewis' and Leep's bounds on the level and sublevel of quaternion algebras are extended to the class of composition algebras. Some simple constructions of composition algebras of known level values are given. In addition, octonion algebras of sublevel 3 and 5 are presented.

On compact realifications of exceptional simple Kantor triple systems

Let A be the realification of the matrix algebra determined by Jordan algebra of hermitian matrices of order three over a complex composition algebra. We define an involutive automorphism on A with a certain action on the triple system obtained from A which give models of simple compact Kantor triple systems. In addition, we give an explicit formula for the canonical trace form and the classification for these triples and their corresponding exceptional real simple Lie algebras. Moreover, we present all realifications of complex exceptional simple Lie algebras as Kantor algebras for a compact simple Kantor triple system defined on a structurable algebra of skew-dimension one

Levels and sublevels of composition algebras

Lewis' and Leep's bounds on the level and sublevel of quaternion algebras are extended to the class of composition algebras. Some simple constructions of composition algebras of known level values are given. In addition, octonion algebras of sublevel 3 and 5 are presented.

Compact realifications of exceptional simple Kantor triple systems defined on tensor products of composition algebras

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.

Models of Compact Simple Kantor Triple Systems Defined on a Class of Structurable Algebras of Skew-Dimension One

Let (A,−): = ℳ(J) be the 2 × 2-matrix algebra determined by Jordan algebra J: = H 3(𝔸) of hermitian 3 × 3-matrices over a real composition algebra 𝔸, where (−) is the standard involution on A. We show that the triple systems B A (x, ∼,z), x,y,z ∈ A, are models of simple compact Kantor triple systems satisfying the condition (A), where B A (x,y,z) is the triple system obtained from the algebra (A,−) and (∼) denotes a certain involution on A. In addition, we obtain an explicit formula for the canonical trace form for the triple systems B A (x, ∼,z).

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.

Irreducible binary (−1, 1)-bimodules over simple finite-dimensional algebras

We prove that an irreducible binary (−1, 1)-bimodule over A is alternative in the following cases: (a) A is a composition algebra over a field of characteristic other than 2 and 3; (b) A is a simple finite-dimensional alternative algebra over a field of characteristic 0.

On composition and absolute-valued algebras

In this note we give a complete description of the composition algebras A over fields of characteristic ≠ 2, 3 in the following cases: if A has an anisotropic norm and x2x = xxx2 for every element; when A has a unitary central idempotent, it satisfies the identity (x2x2)x2 = x2(x2x2), and A is of finite dimension or has anisotropic norm. As a consequence, we obtain the existence, up to an isomorphism, of only seven absolute-valued algebras with a non-zero central idempotent where the last identity holds. This result completes the study of the absolute-valued algebras of this kind that was initiated by El-Mallah and Agawany.We also introduce the class of e-quadratic algebra, which contains the quadratic algebras, but also includes large classes of composition and absolute-valued algebras. Many results on composition, absolute-valued and e-quadratic algebras are shown, and new proofs of some well-known theorems are given.

Division composition algebras through their derivation algebras

Composition algebras are algebras with a nondegenerate quadratic multiplicative form. In this paper we construct five families of division composition algebras and prove that any division composition algebra with nonabelian derivation algebra belongs to one of these families. As a by-product we obtain new examples of real division algebras.

Hall Numbers and the Composition Algebra of the Kronecker Algebra

We present formulas for the structure constants (Hall numbers) of the Hall algebra \(+AFw-mathcal{H}(kK)\) associated to the Kronecker algebra. The formulas which in some cases involve the classical Hall polynomials \(g^{+AFw-lambda}_{(r)+AFw-mu}\) enable us to determine every Hall number. Using again these formulas we construct new PBW-bases with simple structure constants for the composition algebra \(+AFw-mathcal{C}(kK)\), making possible the definition of the generic composition algebra via Hall polynomials.

Composition algebras satisfying certain identities

Even though the class of power associative algebras is huge, only Hurwitz algebras appear in the case of composition algebras, not necessarily unital, over fields of characteristic different from two. Under this assumption and other conditions weaker than power associativity, we study in this note composition algebras of arbitrary dimension. First, a characterization of Hurwitz algebras is given when the ground field has at least five elements. More cases are considered. So, by using conditions on powers of every element, we characterize when a composition algebra over a field of characteristic different from 2 and 3 and with at least 7 elements is Hurwitz or its norm is associative. Similar characterizations are given for composition algebras that are Hurwitz or standard II (respectively Hurwitz or standard III).

An algebraic method for compressing symbolic data tables

Although symbolic data tables summarize huge sets of data they can still become very large in size. This paper proposes a novel technique for compressing a symbolic data table using the recently emerged Compound Term Composition Algebra. One advantage of CTCA is that the closed world hypotheses of its operations can lead to a remarkably high compression ratio. The compacted form apart from having much lower storage space requirements, it allows designing more efficient algorithms for symbolic data analysis.

Geometry over composition algebras: Projective geometry

The purpose of this article is to introduce projective geometry over composition algebras: the analogue of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondence with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. I also give an explicit description of the simply-connected Chevalley group of type E 6 over the integers.

PBW-bases of the twisted generic composition algebras of affine valued quivers

PBW type bases of the twisted generic composition algebras of the affine valued quivers are constructed by using the Frobenius morphism.

Mining the meaningful term conjunctions from materialised faceted taxonomies: algorithms and complexity

A materialised faceted taxonomy is an information source where the objects of interest are indexed according to a faceted taxonomy. This paper shows how from a materialised faceted taxonomy, we can mine an expression of the Compound Term Composition Algebra that specifies exactly those compound terms (conjunctions of terms) that have non-empty interpretation. The mined expressions can be used for encoding in a very compact form (and subsequently reusing), the domain knowledge that is stored in existing materialised faceted taxonomies. A distinctive characteristic of this mining task is that the focus is given on minimising the storage space requirements of the mined set of compound terms. This paper formulates the problem of expression mining, gives several algorithms for expression mining, analyses their computational complexity, provides techniques for optimisation, and discusses several novel applications that now become possible.

THE COMPOSITION ALGEBRA AND THE COMPOSITION MONOID OF THE KRONECKER QUIVER

The Kronecker quiver is considered, and the relations for the specialisation at of the generic composition algebra are given, as well as those for Reineke's composition monoid. As a corollary, it is deduced that the composition monoid is a proper factor of the specialisation of the composition algebra. A normal form is also obtained for the varieties occurring in the composition monoid in terms of Schur roots.

A GENERIC HALL ALGEBRA OF THE KRONECKER ALGEBRA

We construct a generic Hall algebra of the Kronecker algebra and prove that its twisted version is a polynomial algebra in infinitely many variables over the twisted generic composition algebra. The variables are explicitly given as some central elements in the generic Hall algebra. Thus, we obtain generic versions in the Kronecker case of theorems by Hua-Xiao and Sevenhant-Van den Bergh.