R. Parthasarathy

Criteria for the unitarizability of some highest weight modules

For a linear semisimple Lie group we obtain a necessary and sufficient condition for a highest weight module with non-singular infinitesimal character to be unitarizable.

Unitary derived functor modules with small spectrum

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SKEW-PRODUCT FOR GROUP-VALUED EDGE LABELLINGS OF BRATTELI DIAGRAMS

We associate a Cantor dynamical system to a non-properly ordered Bratteli diagram. Group valued edge labellings λ of a Bratteli diagram B give rise to a skew-product Bratteli diagram B(λ) on which the group acts. The quotient by the group action of the associated dynamics can be a nontrivial extension of the dynamics of B. We exhibit a Bratteli diagram for this quotient and con- struct a morphism to B with unique path lifting property. This is shown to be an isomorphism for the dynamics if a property “loops lifting to loops” is satisfied by B(λ) --> B.

Coloring Quasicrystals with Prescribed Symmetries and Frequencies

Abstract. We show how to color the tiles in a heirarchical tiling system so that the resulting system is not only repetitive (i.e., has the local isomorphism property) but has prescribed color symmetries as well.

THE K-GROUP OF SUBSTITUTIONAL SYSTEMS

In another article we associated a dynamical system to a non- properly ordered Bratteli diagram. In this article we describe how to compute the K-group K0 of the dynamical system in terms of the Bratteli diagram. In the case of properly ordered Bratteli diagrams this description coincides with what is already known, namely the so-called dimension group of the Bratteli diagram. The new group defined here is more relevant for non-properly ordered Bratteli diagrams. We use our main result to describe K0 of a substitutional system.

Trace splittings in C*-algebras of tiling systems via colourings

Tiles of a hierarchical tiling system are coloured with given colours. The resulting system implements colour symmetries and prescribed frequencies and is itself a hierarchical system whose prototile types admit an elegant description. The frequencies of occurrence of colours is interpreted using the unique trace on the C*-algebra of the given tiling system and the trace on the C* -algebra of the coloured tiling system.