Speakers

In 1893, Cartan and Engel gave the first realizations of the exceptional simple Lie algebra G(2) as symmetry algebra of various geometric structures. I will report on generalizations of this story to the exceptional simple Lie superalgebras G(3) and F (4), in particular on a supersymmetric extension of the Hilbert–Cartan equation. Joint works with Kruglikov & The.

Exceptionally simple super-PDE

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We construct an infinite family of endofunctors $J^n_d$ on the category of left $A$-modules, where $A$ is a unital associative algebra over a commutative ring $k$, equipped with an exterior algebra $\Omega^\bullet_d$. We prove that these functors generalize the corresponding classical notion of jet functors. The functor $J^n_d$ comes equipped with a natural transformation from the identity functor to itself, which plays the role of the classical prolongation map. This allows us to define the notion of linear differential operator with respect to $\Omega^\bullet_d$. These retain most classical properties of differential operators, and operators such as partial derivatives and connections belong to this class. Moreover, we construct a functor of quantum symmetric forms $S^n_d$ associated to $\Omega^\bullet_d$, and proceed to introduce the corresponding noncommutative analogue of the Spencer $\delta$-complex. We give necessary and sufficient conditions under which the jet functor $J^n_d$ satisfies the jet exact sequence, $0\to S^n_d \to J^n_d \to J^{n−1}_d \to 0$. This involves imposing mild homological conditions on the exterior algebra, in particular on the Spencer cohomology $H^{\bullet,2}$. This is a joint work with K. Flood and H- Winther.

Jet functors in noncommutative geometry

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In this talk we show how the C*-completions of the Drinfeld--Jimbo quantum flag manifolds can be realised as graph C*-algebras. We begin by recalling how to construct a C*-algebra from a directed graph, how to read the K-theory groups of the C*-algebra directly from the graph, and how to see its ideal structure. We then recall how to compute the primitive ideal space by using Dijkhuizen and Stokmann's description of a complete set of irreducible *-representations. Finally, we show how to construct a graph directly from the Weyl group of the associated Lie algebra, and show that we recover known isomorphisms between the C*-algebras of quantum flag manifolds, as well as determining surprising new isomorphisms. (Joint work with Tomasz Brzeziński, Ulrich Krähmer, and Karen Strung)

Connections on vector bundles are a fundamental tool in classical differential geometry. When we generalize their definition to noncommutative differential geometry, we are led to consider bimodule connections. In this talk I will present a construction of bimodule connections for line bundles over a large class of quantum homogeneous spaces, namely the quantum flag manifolds. Generalizing the work of Beggs and Majid for the Podles sphere, we realize bimodule connections as associated to a principal connection for the Heckenberger-Kolb calculus . Time allowing, I will review explicit presentations of the associated bimodule maps first in terms of generalised quantum determinants and then in terms of Takeuchi's categorical equivalence for relative Hopf modules.

Bimodule connections for line bundles over irreducible quantum flag manifolds

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Quantum groups – in particular, quantized universal enveloping algebras, or "QUEA’s" – over a symmetrizable Kac-Moody algebra were initially introduced as depending on just one “continuous” parameter, both in a "formal" and in a "polynomial" version. Later on, multiparameter QUEA’s have been introduced, with additional “discrete” parameters that affect only either the coalgebra structure or the algebra structure. In both cases, these multiparameter QUEA's can be realized as (Hopf) deformations – by twist or by 2-cocycle – of Drinfeld's celebrated QUEA, which is *uniparameter* instead. In this talk I will introduce a newly minted family of (formal) multiparameter QUEA’s that encompasses and generalizes the previous ones: moreover, this family is also stable with respect to both deformations by twists and by 2-cocycles (of a special, "toral" type). Taking semiclassical limits, these new multiparameter QUEA’s give rise to a new family of multiparameter Lie bialgebras, that in turn is stable under both deformations by twists and by 2-cocycles (again of "toral" type), in the sense of Lie bialgebras. All of this is a joint work with Gastón Andrés García, freely available as preprint arXiv:2203.11023 (2022).

A unifying approach to multiparameter quantum groups

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Atiyah sequences of braided Lie algebras and their splittings

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