Congruences in right quasigroups and general extensions

Abstract

The Schur multipliers of reflection groups were first constructed by I. Ihara and T. Yokonuma. Later, R. B. Howlett gave a more unified, but less concrete, construction for arbitrary Coxeter groups. In this paper, Clifford algebras are used to give a more unified approach than that originally used by I. Ihara and T. Yokonuma.     

Structure globale des extensions regulieres galoisiennes

A natural characterization of known types of inverse transversal leads to the introduction of several new types, which we characterize in a similar natural way. Relations between the various types produce a general classification. We also determine for which types of inverse transversal considered the congruence extension property holds.     

Limits of rank 4 Azumaya algebras and applications to desingularization

It is shown that the schematic image of the scheme of Azumaya algebra structures on a vector bundle of rank 4 over any base scheme is separated, of finite type, smooth of relative dimension 13 and geometrically irreducible over that base and that this construction base-changes well. This fully generalizes Seshadri’s theorem in [16] that the variety of specializations of (2 x 2)-matrix algebras is smooth in characteristic ≠ 2. As an application, a construction of Seshadri in [16] is shown in a characteristic-free way to desingularize the moduli space of rank 2 even degree semi-stable vector bundles on a complete curve. As another application, a construction of Nori over ℤ (Appendix, [16]) is extended to the case of a normal domain which is a universally Japanese (Nagata) ring and is shown to desingularize the Artin moduli space [1] of invariants of several matrices in rank 2. This desingularization is shown to have a good specialization property if the Artin moduli space has geometrically reduced fibers — for example this happens over ℤ. Essential use is made of Kneser’s concept [8] of ‘semi-regular quadratic module’. For any free quadratic module of odd rank, a formula linking the half-discriminant and the values of the quadratic form on its radical is derived.     

Une Neutralisation Explicite de L'algèbre de Weyl Quantique Complétée

Soit p un nombre premier. Nous établissons l'existence de neutralisations de divers complétés de l'algèbre de Weyl quantique spécialisée en une racine de l'unité primitive d'ordre p (qui est “génériquement” une algèbre d'Azumaya) et donnons en particulier un énoncé de neutralisation explicite relevant celui construit en caractéristique p dans [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].

Let p be a prime number. We establish the existence of neutralizations of various completions of the quantum Weyl algebra specialized at a primitive root of unity of prime order p (which is “generically” an Azumaya algebra) and, in particular, we give a statement of explicit neutralization similar to the one built in characteristic p in [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].     

H-separable Hopf Galois Extensions and Azumaya Algebra

1 IntroductionLet A/R be a ring extension with the common identity 1. A/R is said to be separable if theA-bimodule homomorphism of A @R A onto A defined by a @ 5-a6 splits. A separableextension over a non-commutative ring generalizes that over a commutative ring which wasdiscussed in [1]. Hirata introduced anOther kind of separable extensions called H-separabeones (see [2]). A/R is said to be H-separable if A @R A is isomorphic as an A-bimoduleto a direct sumrnand of A". riom {2, Theor…     

Geometric Algebra for Subspace Operations

The set theory relations , \,,, and have corollaries in subspace relations. Geometric algebra is introduced as a useful framework to explore these subspace operations. The relations , \, and are easily subsumed by geometric algebra for Euclidean metrics. A short computation shows that the meet () and join () are resolved in a projection operator representation with the aid of one additional product beyond the standard geometric algebra products. The result is that the join can be computed even when the subspaces have a common factor, and the meet can be computed without knowing the join. All of the operations can be defined and computed in any signature (including degenerate signatures) by transforming the problem to an analogous problem in a different algebra through a transformation induced by a linear invertible function (a LIFT to a different algebra). The new results, as well as the techniques by which we reach them, add to the tools available for subspace computations.     

A Group of Involutions Characterizing a Clifford Algebra of Type {\mathcal{C}}\,{\mathcal{L}}_{0,3}

In this note, a group characterization for Clifford algebras of type is given.     

Obstructions to clifford system extensions of algebras

In this paper we do phrase the obstruction for realization of a generalized group character, and then we give a classification of Clifford systems in terms of suitable low-dimensional cohomology groups.     

Primitive axial algebras are of Jordan type

The notion of axial algebra is closely related to 3-transposition groups, the Monster group and vertex operator algebras. In this work we continue our previous works and complete the proof that all algebras generated by a set of primitive axes not necessarily of the same type (see the definition in the body of the paper), are primitive axial algebras of Jordan type.