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1
Content available remote Multiplicity formulas for representations of transformation groupoids
Giżycki A., Pysiak L.
Demonstratio Mathematica
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2017
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Vol. 50, nr 1
42--50
EN
We study the representations of transitive transformation groupoids with the aim of generalizing the Mackey theory. Using the Mackey theory and a bijective correspondence between the imprimitivity systems and the representations of a transformation groupoid we derive the irreducibility theory. Then we derive the direct sum decomposition for representations of a groupoid together with the formula for the multiplicity of subrepresentations. We discuss a physical interpretation of this formula. Finally, we prove the claim analogous to the Peter–Weyl theorem for a noncompact transformation groupoid. We show that the representation theory of a transitive transformation groupoids is closely related to the representation theory of a compact groups.
2
Content available remote Semidirect product of groupoids and associated algebras
Pysiak L., Eckstein M., Heller M., Sasin W.
Demonstratio Mathematica
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2014
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Vol. 47, nr 2
289--299
EN
One of the pressing problems in mathematical physics is to find a generalized Poincaré symmetry that could be applied to nonflat space-times. As a step in this direction, we define the semidirect product of groupoids Γ0 x Γ1 and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid Γ1 and prove that it is isomorphic to the convolutive algebra of the groupoid Γ0 x Γ1. We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. An important example is the Poincaré groupoid defined as the semidirect product of the subgroupoid of generalized Lorentz transformations and the subgroupoid of generalized translations.
3
Content available remote Imprimitivity theorem for groupoid representations
Pysiak L.
Demonstratio Mathematica
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2011
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Vol. 44, nr 1
29-48
EN
We define and investigate the concept of the groupoid representation in- duced by a representation of the isotropy subgroupoid. Groupoids in question are locally compact transitive topological groupoids. We formulate and prove the imprimitivity theorem for such representations which is a generalization of the classical Mackey's theorem known from the theory of group representations.
4
Content available remote Hopf-Sikorski algebras
Heller M., Odrzygóźdź Z., Pysiak L., Sasin W.
Demonstratio Mathematica
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2011
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Vol. 44, nr 2
213-221
EN
The dual category with respect to the category of differential groups is defined and investigated. The objects of this category are algebras, called Hopf-Sikorski (H-S) algebras, the axioms of which combine the axioms of Sikorski's algebras with modified axiomas of Hopf algebras. Morphisms of this category are structural mappings corresponding to Hopf algebras that are smooth in the sense of Sikorski. As an example, we discuss the H-S algebra of the Lorentz group.
5
Content available remote Inner geometry of random operators
Heller M., Pysiak L., Sasin W.
Demonstratio Mathematica
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2006
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Vol. 39, nr 4
971-978
EN
If V is a foliated manifold, there exists a von Neumann algebra M associated with V. We consider the case when V is a transformation groupoid gamma and the von Neumann algebra M associated with gamma is a noncommutative algebra of random operators. We show that M is generated by a functional algebra A defined on the groupoid gamma with a noncommutative convolution as multiplication, and develop the differential geometry (metric, connection and curvature) based on inner derivations of the algebra A.
6
Content available remote Quantum groupoids of the finite type and quantization on orbit spaces
Heller M., Odrzygóźdź Z., Pysiak L., Sasin W.
Demonstratio Mathematica
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2004
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Vol. 37, nr 3
671--678
EN
We show that the Hopf algebra on a transformation groupoid F = E x G where G is a finite group acting on the total space of a principal fibre boundle over M = E/G, is the cross product of the algebras C°°(E) and CG. We study duality properties of this algebra, and consider quantization on orbit spaces program in this context.
7
Content available remote Groupoids, their representations and imprimitivity systems
Pysiak L.
Demonstratio Mathematica
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2004
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Vol. 37, nr 3
661--670
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