Formally real fields,pythagorean fields,C-fields andW-groups

Supported in part by the National Science Foundation and the Natural Sciences and Engineering Research Council of Canada     

Generating the alternating group by cyclic triples

It is shown that a collection of circular permutations of length three on an n-set generates the alternating group A_n if and only if the associated graph is connected. It follows that [$\text{1}\text{2}\text{n}$] circular permutations of length three may generateA_n.     

The Schur group of cyclotomic fields

We compute the Schur group of the cyclotomic fields Q(?_m) and real quadratic fields $\text{Q(d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ where d is a product of an even number of primes congruent to three modulo four. Some results are also given about the Schur group of certain subfields of cyclotomic fields.     

Weak additivity of metric pythagorean orthogonality

It is well known that the property of additivity of pythagorean orthogonality characterizes real inner product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes real inner product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.     

The rank of group of cyclotomic units in abelian fields

A formula about the rank of group of cyclotomic units in abelian fields is established. From that formula, a series of equivalent conditions for independence of the system of cyclotomic units in abelian fields is stated and proved. For the particular case of cyclotomic fields, further properties of the rank are researched.     

The fundamental group of period domains over finite fields

We determine the fundamental group of period domains over finite fields. This answers a question of M. Rapoport raised in [M. Rapoport, Period domains over finite and local fields, in: Proc. Sympos. Pure Math., vol. 62, part 1, 1997, pp. 361-381].     

Number fields unramified away from 2

Consider the set of number fields unramified away from 2, i.e., unramified outside {2,∞}. We show that there do not exist any such fields of degrees 9 through 15. As a consequence, the following simple groups are ruled out for being the Galois group of an extension which is unramified away from 2: Mathieu groups M₁₁ and M₁₂, PSL(3,3), and alternating groups Aj for 8<j<16 (values j?8 were previously known).     

Number of generators of a group

Mathematical Notes -     

The classification of Leonard triples of QRacah type

Let $\mathbb{K}$ denote an algebraically closed field. Let V denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations in $\mathrm{End}(V)$ such that for each of these transformations there exists a basis of V with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the Leonard triples of QRacah type up to isomorphism. We show that any Leonard triple of QRacah type satisfies the ${\mathbb{Z}}_3$-symmetric Askey-Wilson relations.     

The classification of Leonard triples of Racah type

Let K$\mathbb{K}$ denote an algebraically closed field of characteristic zero. Let V   denote a vector space over K$\mathbb{K}$ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A  , A^?$A^{?}$, A^ε$A^{\epsilon }$ in End(V)$\mathrm{End}(V)$ such that for each B∈{A,A^?,A^ε}$B\in \{A,A^{?},A^{\epsilon }\}$ there exists a basis for V with respect to which the matrix representing B   is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper we define a family of Leonard triples said to have Racah type and classify them up to isomorphism. Moreover, we show that each of them satisfies the _Z3${\mathbb{Z}}_3$-symmetric Askey–Wilson relations. As an application, we construct all Leonard triples that have Racah type from the universal enveloping algebra U(_sl2)$U({\mathit{sl}}_2)$.     

A couple of triples

The standard contravariant adjunction between TOP (the category of topological spaces) and LAT (the category of distributive lattices) induces a triple Λ on LAT and a triple Σ on TOP. We show that the category LAT^Λ of Λ-algebras is just the category of frames, and describe the category TOP^Σ of Σ-algebras as a subcategory of TOP.