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1

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

Porings and poring algebras

Jun Y., Kim H., Neggers J.

Demonstratio Mathematica

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2006

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Vol. 39, nr 2

267-276

EN

In this paper we define an algebra of type (2,2,0) associated with posets called a poring and we study several properties of porings and the linear algebras having such porings as their bases. In particular, we show that if (P; *, .) is a standard poring then the distributive law (x . y) * z = (x * z) . (y * z) holds iff the poset P (or the poset X) is (C2 +1)-free.

3

Left symmetric left distributive magmas and hypersubstitutions

Vanzurova A.

Demonstratio Mathematica

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2006

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Vol. 39, nr 3

491-506

EN

Our aim is to generalize results reached in [A-D 98] and [V 04]. In [A-D 98], normal forms for terms with respect to the variety SIE of right symmetric idempotent entropic magmas are used to derive multiplication in the magma of normal form hypersubstitutions with respect to SIE, the monoid of SIE-proper normal form hypersubstitutions is found, and hyperidentities are discussed. In [V 04], a similar project is solved for the variety SID of left symmetric left distributive idempotent magmas (in which the variety dual to SIE is contained as a subvariety). Droppping idempotency we obtain a generalization, the variety SD of left symmetric left distributive magmas. We use again (naturally arising) normal forms for terms in SD to study the magma of SD-normal form hypersubstitutions, its multiplication (with six idempotents), and describe the monoid of SD-proper normal form hypersubstitutions. Comparison with the previous cases might be interesting.

4

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.

5

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.

6

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