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A groupoid formulation of the Baire Category Theorem

Jonathan Brown, Lisa Orloff Clark (2014)

Fundamenta Mathematicae

We prove that the Baire Category Theorem is equivalent to the following: Let G be a topological groupoid such that the unit space is a complete metric space, and there is a countable cover of G by neighbourhood bisections. If G is effective, then G is topologically principal.

An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds

Hamid-Reza Fanaï, Atefeh Hasan-Zadeh (2019)

Mathematica Bohemica

We study a problem of isometric compact 2-step nilmanifolds M / Γ using some information on their geodesic flows, where M is a simply connected 2-step nilpotent Lie group with a left invariant metric and Γ is a cocompact discrete subgroup of isometries of M . Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization...

An introduction to loopoids

Janusz Grabowski (2016)

Commentationes Mathematicae Universitatis Carolinae

We discuss a concept of loopoid as a non-associative generalization of Brandt groupoid. We introduce and study also an interesting class of more general objects which we call semiloopoids. A differential version of loopoids is intended as a framework for Lagrangian discrete mechanics.

Bol loop actions

Larissa V. Sbitneva (2000)

Commentationes Mathematicae Universitatis Carolinae

The notions of left Bol and Bol-Bruck actions are introduced. A purely algebraic analogue of a Nono family (Lie triple family), the so called Sabinin-Nono family, is given. It is shown that any Sabinin-Nono family is a left Bol-Bruck action. Finally it is proved that any local Nono family is a local left Bol-Bruck action. On general matters see [L.V. Sabinin 91, 99].

From double Lie groups to quantum groups

Piotr Stachura (2005)

Fundamenta Mathematicae

It is shown, using geometric methods, that there is a C*-algebraic quantum group related to any double Lie group (also known as a matched pair of Lie groups or a bicrossproduct Lie group). An algebra underlying this quantum group is the algebra of a differential groupoid naturally associated with the double Lie group.

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