Quantum groups

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Any progress in this direction will have a major long-term impact on the mathematical community, and on the scientific environment in the UK and internationally. We plan to attack the problem by combining a thorough study of a very diverse set of representations of these exotic algebras together with the development of new techniques to treat quantum superalgebras, and to derive from this combination a universal mathematical formulation which captures the common features of these representations and generalizes them. From this formulation, we plan to derive new results on quantum groups and apply them to the problem of non-ultralocality in integrable systems, following a top-down approach.

The tools will primarily consist of the representation theory of Lie algebras and superalgebras and the technology of finite and infinite dimensional Hopf algebras, Lie bialgebras and their associated symplectic structures. The intradisciplinary character of the project, combining ideas and techniques from different areas of mathematics, will lead to new results across a broad range of topics, from group theory to geometry Hamiltonian structures , topology knot invariants, Grassmannian manifolds and mathematical physics. Exotic quantum groups, Lie superalgebras and integrable systems.

Torrielli, Dr A. University of Surrey.

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First Grant - Revised No relevance to Underpinning Sectors. Panel Date.

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Mathematics Prioritisation Panel Meeting December Summary on Grant Application Form. In recent years, motivated by the study of a particular class of integrable systems, new remarkable mathematical structures have been discovered which had not been investigated before. Key Findings. A Ayala, D.. Factorization homology of topological manifolds. Factorization homology of stratified spaces. Selecta Mathematica. Arinkin, D.. Singular support of coherent sheaves and the geometric Langlands conjecture. Combinatorial quantization of the Hamiltonian Chern-Simons theory.

Alekseev, A.. Quasi-Poisson manifolds. Symplectic structure of the moduli space of flat connection on a Riemann surface - Alekseev, A. Andersen, H. Representations of quantum algebras. Alekseev, A.

EMS - European Mathematical Society Publishing House

Representation theory of Chern-Simons observables. Duke Math. Audin, M.. Lectures on gauge theory and integrable systems. Hurtubise, F. Lalonde, G. Sabidussi eds. Gauge Theory and Symplectic Geometry, vol.

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Beilinson, A.. Chiral algebras, vol. Blumberg, A. A universal characterization of higher algebraic k-theory. Brochier, A.. Fourier transform for quantum D-modules via the punctured torus mapping class group. Quantum Topol. Dualizability of braided tensor categories. In preparation. Bakalov, B.. Lectures on tensor categories and modular functors, vol. Backelin, E.. Quantum flag varieties, equivariant quantum D-modules, and localization of quantum groups.

Localization for quantum groups at a root of unity. Blackwell, R.. Two-dimensional monad theory. Pure Appl.


Quantum Groups

Bruguires, A.. Exact sequences of tensor categories. Baumann, P.. Classification of bicovariant differential calculi on quantum groups a representation-theoretic approach. Ben-Zvi, D.. Quantum character varieties and braided module categories. The character theory of a complex group. Calaque, D..

On representations of quantum groups and Cherednik algebras

Universal KZB equations: The elliptic case. Tschinkel, Y. Zarhin eds. Algebra, Arithmetic, and Geometry, vol. Costello, K..

Kazhdan's Property T for Discrete Quantum Groups

Factorization algebras in perturbative quantum field theory Preprint, Available at. Crane, L..

State-sum invariants of 4-manifolds. Knot Theory Ramifications.

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  7. A guide to quantum groups - Chari, V. Cambridge, UK: Univ. Factorization homology as a fully extended topological field theory.