Final Report Summary - PROBQUANTGROUPS (Probability and quantum groups)
The PROBQUANTGROUPS project was placed at the intersection of functional analysis, probability theory, quantum algebra and mathematical physics.
The main results obtained during the fellowship were related to the analysis of quantum symmetries, represented via quantum groups and their actions, with a special focus on the quantum probabilistic approach. Below, we summarise the main concrete scientific achievements obtained in the 6 papers written and published within the period of the Fellowship:
- New examples of quantum symmetry groups have been presented, with their categories of representations computed in terms of certain categories of partitions and given interpretations in the language of free probability.
- Connections between idempotent states and inner faithfulness of representations of algebras of functions on compact quantum groups, opening a new path to solving an important problem of verifying inner linearity of quantum permutation groups.
- A study of various notions of quantum homogenous spaces understood as 'rectangular matrix parts' of certain free matrix quantum groups was initiated and developed.
- Several possible definitions of closed quantum subgroups of locally compact quantum groups available in the literature were compared, identified in several cases and studied in depth.
- The class of compact quantum groups not allowing non-trivial square roots of the Haar states, generalising the class of classical groups investigated by Diaconis and Shahshahani, was identified and characterised in terms of the representation theory.
The above outcomes can be viewed as extending the longstanding fruitful interaction between classical probability, geometry and algebraic structures to the non-commutative or quantum context. Apart from these achievements, several other results have been obtained and new directions of research unveiled; these form the content of 5 further papers accepted for publication, three preprints and several articles that are currently being prepared. This means in particular that all the research objectives listed in the application have been at least partially achieved, with certain natural differences resulting from the general scientific development of the areas of the project within the time that passed since the application.
All the outcomes described above, although of purely mathematical nature, and immediately of interest to the mathematical community, offer also long term impact on mathematical physics, and increase the competitiveness of the European Union (EU) in the area of fundamental sciences.
The excellent working conditions offered by the fellowship allowed the fellow to continue and strengthen the previously existing collaborations, and also to develop new scientific contacts, both in Warsaw, where the fellowship was held, and in other European and other countries (Canada, Finland, France).
The experience and results obtained within the period of the fellowship have aided fellow in obtaining a long-term academic post, offered jointly by the Mathematical Institute of the Polish Academy of Sciences and the University of Warsaw. This assures the long distance impact of the fellowship and the continuation of the related research beyond the period of the direct EU funding.
The main results obtained during the fellowship were related to the analysis of quantum symmetries, represented via quantum groups and their actions, with a special focus on the quantum probabilistic approach. Below, we summarise the main concrete scientific achievements obtained in the 6 papers written and published within the period of the Fellowship:
- New examples of quantum symmetry groups have been presented, with their categories of representations computed in terms of certain categories of partitions and given interpretations in the language of free probability.
- Connections between idempotent states and inner faithfulness of representations of algebras of functions on compact quantum groups, opening a new path to solving an important problem of verifying inner linearity of quantum permutation groups.
- A study of various notions of quantum homogenous spaces understood as 'rectangular matrix parts' of certain free matrix quantum groups was initiated and developed.
- Several possible definitions of closed quantum subgroups of locally compact quantum groups available in the literature were compared, identified in several cases and studied in depth.
- The class of compact quantum groups not allowing non-trivial square roots of the Haar states, generalising the class of classical groups investigated by Diaconis and Shahshahani, was identified and characterised in terms of the representation theory.
The above outcomes can be viewed as extending the longstanding fruitful interaction between classical probability, geometry and algebraic structures to the non-commutative or quantum context. Apart from these achievements, several other results have been obtained and new directions of research unveiled; these form the content of 5 further papers accepted for publication, three preprints and several articles that are currently being prepared. This means in particular that all the research objectives listed in the application have been at least partially achieved, with certain natural differences resulting from the general scientific development of the areas of the project within the time that passed since the application.
All the outcomes described above, although of purely mathematical nature, and immediately of interest to the mathematical community, offer also long term impact on mathematical physics, and increase the competitiveness of the European Union (EU) in the area of fundamental sciences.
The excellent working conditions offered by the fellowship allowed the fellow to continue and strengthen the previously existing collaborations, and also to develop new scientific contacts, both in Warsaw, where the fellowship was held, and in other European and other countries (Canada, Finland, France).
The experience and results obtained within the period of the fellowship have aided fellow in obtaining a long-term academic post, offered jointly by the Mathematical Institute of the Polish Academy of Sciences and the University of Warsaw. This assures the long distance impact of the fellowship and the continuation of the related research beyond the period of the direct EU funding.