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Classification of spherical varieties

Paolo Bravi (2010)

Les cours du CIRM

We give a short introduction to the problem of classification of spherical varieties, by presenting the Luna conjecture about the classification of wonderful varieties and illustrating some of the related currently known results.

Classification of strict wonderful varieties

Paolo Bravi, Stéphanie Cupit-Foutou (2010)

Annales de l’institut Fourier

In the setting of strict wonderful varieties we prove Luna’s conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that primitive strict wonderful varieties are mostly obtained from symmetric spaces, spherical nilpotent orbits and model spaces. To make the paper as self-contained as possible, we also gather some known results on these families and more generally on wonderful varieties.

Classification results in quasigroup and loop theory via a combination of automated reasoning tools

Volker Sorge, Simon Colton, Roy McCasland, Andreas Meier (2008)

Commentationes Mathematicae Universitatis Carolinae

We present some novel classification results in quasigroup and loop theory. For quasigroups up to size 5 and loops up to size 7, we describe a unique property which determines the isomorphism (and in the case of loops, the isotopism) class for any example. These invariant properties were generated using a variety of automated techniques --- including machine learning and computer algebra --- which we present here. Moreover, each result has been automatically verified, again using a variety of techniques...

Clifford algebras, Möbius transformations, Vahlen matrices, and B -loops

Jimmie Lawson (2010)

Commentationes Mathematicae Universitatis Carolinae

In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball B in n -dimensional euclidean space n . One notable achievement is a compact, convenient formula for the Möbius loop operation a * b = ( a + b ) ( 1 - a b ) - 1 , where the operations on the right are those arising from the Clifford algebra (a formula comparable to ( w + z ) ( 1 + w ¯ z ) - 1 for the Möbius loop multiplication...

Clifford congruences on generalized quasi-orthodox GV-semigroups

Sunil K. Maity (2013)

Discussiones Mathematicae - General Algebra and Applications

A semigroup S is said to be completely π-regular if for any a ∈ S there exists a positive integer n such that aⁿ is completely regular. A completely π-regular semigroup S is said to be a GV-semigroup if all the regular elements of S are completely regular. The present paper is devoted to the study of generalized quasi-orthodox GV-semigroups and least Clifford congruences on them.

Clifford semifields

Mridul K. Sen, Sunil K. Maity, Kar-Ping Shum (2004)

Discussiones Mathematicae - General Algebra and Applications

It is well known that a semigroup S is a Clifford semigroup if and only if S is a strong semilattice of groups. We have recently extended this important result from semigroups to semirings by showing that a semiring S is a Clifford semiring if and only if S is a strong distributive lattice of skew-rings. In this paper, we introduce the notions of Clifford semidomain and Clifford semifield. Some structure theorems for these semirings are obtained.

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