### Speakers/Abstracts

### Seminars

- Paolo Aschieri: Noncommutative Principal Bundles and their gauge group via Drinfeld twist

Drinfeld twist deformation theory of modules and algebras that carry a representation of a Hopf Algebra H can be extended in order to canonically deform principal G-bundles. Twisting the structure group we obtain principal bundles with noncommutative fiber and where the structure group is a quantum group. Twisting the automorphism group we further obtain a noncommutative base space. In the first case an explicit description of the gauge group (vertical automorphisms) is presented. - Ruggero Bandiera: Eulerian idempotent, pre-Lie logarithm and combinatorics of trees

The aim of this talk is to bring together the three objects in the title. Recall that, given a Lie algebra g, the Eulerian idempotent is a canonical projector from the enveloping algebra U(g) to g. The Baker-Campbell-Hausdorff product and the Magnus expansion can both be expressed in terms of the Eulerian idempotent, which makes it interesting to look for explicit formulas for the latter. We show how to reduce the computation of the Eulerian idempotent to the computation of a logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The problem of finding formulas for the pre-Lie logarithm, which is interesting in its own right, with connections to operad theory, numerical analysis and renormalization, is addressed using techniques inspired by umbral calculus. As a consequence of our analysis, we find formulas both for the Eulerian idempotent and the pre-Lie logarithm in terms of combinatorics of trees. Based on joint work with Florian Schaetz. - Jill Ecker: The Vanishing of Cohomology Groups of the
Witt and the Virasoro Algebra

The aim of our work is to study the low-dimensional cohomology groups of the Witt and the Virasoro algebra from a purely algebraic viewpoint. The talk starts with a brief introduction of the Witt algebra and its central extension, the Virasoro algebra. In a second step, the Chevalley-Eilenberg cohomology of Lie algebras is described, with a particular focus on low-dimensional cohomology groups with values in the adjoint module. We will illustrate our algebraic techniques on the proof of the vanishing of the first cohomology group with values in the adjoint module of the Witt algebra. We will finish by presenting the outline of the proof of the same result for the third cohomology group, accompanied by some concrete computations. The proof is a generalization of the one developed by Schlichenmaier to prove the vanishing of the second cohomology group. This is joint work with Martin Schlichenmaier. - Niek de Kleijn: Quantizing Infinitesimal Actions through Formality

Constructing deformation quantizations of Poisson manifolds M may be done in various ways. When the Poisson manifold allows an infinitesimal Poisson action of a triangular Lie algebra (g,r) we can find deformation quantizations through the universal deformation formulas of Drinfeld. One uses essentially a deformation of U(g) as a Hopf algebra given by a Drinfeld Twist J. Another more general way is to obtain explicit*L*-formality maps for the polydifferential operators on M. We apply the formality in the case of Lie algebroids to the case where we have an infinitesimal Poisson action to obtain a generalization of the situation concerning Drinfeld twists. In particular we obtain a generalized notion of quantum action that yields universal deformation formulas._{∞} - Ulrich Krähmer: Cyclic vs mixed homology

Mixed (or duchain) complexes are chain complexes which are simultaneously cochain complexes. Standard examples are the Hochschild chain complex of an associative algebra, the differential forms on a Poisson manifold, but also more classically on an oriented Riemannian manifold. In this talk I will discuss homological invariants that one can define in this situaiton and how they are related (based on joint work with Dylan Madden). - Laurent La Fuente Gravy: Moment maps and closed Fedosov's star products

I will describe a moment map on the space of symplecic connections on a given closed symplectic manifold. The value of this moment map at a symplectic connection is contained in the trace functional of the Fedosov's star pro duct attached to this connection. Moreover, this Fedosov's star pro duct can only be closed when the symplectic connection lies in the vanishing set of the moment map. Considering closed Kähler manifolds and working on Kähler potentials, I will show that the problem of finding zero es of the moment map is an elliptic partial differential equation. On complex tori and complex projective spaces, I will show that part of the zero set of the moment map has the structure of a finite dimensional manifold. I will also present an obstruction to the existence of zero es of the moment map, which gives an obstruction to the closedness of the Fedosov's star pro duct attached to the considered Kähler data. - Hsuan-Yi Liao: Formality theorem for g-manifolds

To a g-manifold, a smooth manifold with a Lie algebra action, one can associate two dglas which play roles similar to the spaces of polyvector fields and polydifferential operators. We establish the formality theorem for g-manifolds, i.e., an*L*quasi-isomorphism from the dgla analogous to polyvector fields to the dgla analogous to polydifferential operators. As an application, we obtain a Kontsevich–Duflo type theorem for g-manifolds. The parallel theorems for other cases will also be mentioned briefly._{∞} - Madeleine Jotz Lean: Split Lie 2-algebroids and matched pairs of 2-representations

A matched pair of Lie algebroid representations is equivalent to two seemingly different objects; its bicrossproduct Lie algebroid and its double Lie algebroid. In this talk I define matched pairs of 2-term Lie algebroid representations up to homotpy (2-representations for short). I prove that the bicrossproduct of a matched pair of 2-representations is a split Lie 2-algebroid, and I explain the geometric insight -- in the context of double Lie algebroids and VB-Lie bialgebroids -- at the origin of this algebraic construction. This explains in particular how the double of a matched pair of representations can be seen as the geometrisation of the bicrossproduct Lie algebroid that matched pair. I sketch more generally an equivalence of Lie 2-algebroids with VB-Courant algebroids, and, if time permits, I explain how the cotangent double of a Lie bialgebroid can be interpreted as a geometrisation of its bicrossproduct Courant algebroid. - Elena Martinengo: Kaledin class and formality of the moduli space of sheaves on K3s.

In 2007 Kaledin proved a geometric theorem of formality in familes using as major tool a class that he associated to a differential graded algebra (or in general to an A-infinity algebra). In this talk I will explain how this "Kaledin" class is defined, in which part of the Hochschild cohomology it lives and why its vanishing controls the formality. As an application, I will sketch a proof of the conjecture formulated by Kaledin and Lehn in 2007 about the singularities of the moduli space of sheaves on a K3 surface. This last part is a work in progress with Manfred Lehn. - Sergei Merkulov: An explicit two step quantization of Poisson structures and Lie bialgebras

We discuss a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp. Lie bialgebra) structure. We show explicit transcendental formulae for this correspondence. In the second step one deformation quantizes a quantizable Poisson (resp. Lie bialgebra) structure, also via explicit transcendental formulae. In the Poisson case the first step is the most non-trivial one and requires a choice of an associator while the second step quantization is essentially unique, it is independent of a choice of an associator and can be done by a trivial induction. We conjecture that similar statements hold true in the case of Lie bialgebras. The main new result is a surprisingly simple explicit universal formula (which uses only smooth differential forms) for universal quantizations of finite-dimensional Lie bialgebras. The talk is based on a joint work with Thomas Willwacher. - Hessel Posthuma: Cyclic homology, Lie algebroids and the adjoint representation

Lie algebroids are joint generalizations of Lie algebras and tangent bundles. In this talk I will report on joint work with Arie Blom computing the cyclic homology of the universal enveloping algebra of a Lie algebroid, recovering well-known results of Kassel and Wodzicki in the special cases mentioned above. We will show how the adjoint representation (up to homotopy) plays a crucial role in these computations. - Jonas Schnitzer: Generalized Geometry in odd Dimensions

After a short introduction to Jacobi manifolds and their role in Poisson geometry, we introduce the notion of generalized contact manifolds. As well as contact manifolds are the odd dimensional correspondance of symplectic manifolds, generalized contact manifolds are the odd dimensional counterpart of generalized complex manifolds. Using a version of the symplectization trick, defining a symplectic manifold out of a contact manifold, we can define a generalized complex structure out of the generalized contact manifold. At the end, we want to see that these structures are build up out of already known ones, which are symplectic, contact and generalized complex structures. This is a joint work with Luca Vitagliano. - Matthias Schötz: Non-formal Deformation Quantisation and (abstract) O*-Algebras

There has been some effort in constructing and examining examples of non-formal deformation quantisations besides the well-known framework of deformation of C*-algebras. On the upside, this allows to construct more examples and examples with certain properties that are prohibited in the C* case, e.g. algebras that contain position and momentum operators Q and P with canonical commutation relations [ Q, P ] = iħ, and not only, say, their exponentials. On the downside, the resulting examples are in many cases not even locally multiplicatively convex and thus their properties are not very well understood. One way to examine such *-algebras could be to develop a notion of abstract O*-algebras, which treat *-algebras of unbounded operators in a representation-independent way, analogously to how C*- and W*-algebras abstract the notion of *-algebras of bounded operators.

In this talk I am going to present two recent examples of non-formal and non-C* quantisations of *-algebras (on flat and hyperbolic space) and discuss some of their properties as abstract O*-algebras. - Luca Vitagliano: The deformation
*L*-algebroid of a foliation_{∞}

We show that the deformation complex of a foliation*F*is equipped with an*L*-algebroid structure_{∞}*L(F)*which is unique up to isomorphisms. Time permitting, we will also discuss a possible definition of the universal enveloping*A*-algebra of_{∞}*L(F)*. - Stefan Waldmann: Convergence of Formal Star Products

In this talk I will present some general ideas on questions of convergence of formal star products together with several classes of examples, complementing the talk of Matthias Schötz. I will argue that the framework of locally convex algebras is suitable to cover several interesting and physically relevant examples. It can be seen as bridge between the formal deformations and $C^*$-algebraic version. - Thomas Weber: Equivariant Morita Equivalence and Twist Star Products

The notion of Drinfel’d twist gives rise to deformations of Hopf algebras and their module algebras at once. In the case of a universal enveloping algebra of a Lie algebra acting on a manifold via derivations, a twist induces a deformation quantization of the manifold. For example, the Moyal-Weyl star product emerges in this way. While it is well-known that every Poisson manifold admits a deformation quantization, it is not clear which deformations arise by this twisting procedure. One can exclude the symplectic 2-sphere and the higher symplectic Riemann surfaces to inherit such twist star products by some simple geometrical and topological arguments, but besides that not much is known. In this talk we will connect the existence of twist star products to the existence of equivariant line bundles with non-trivial Chern class. To this aim, we introduce the theory of equivariant Morita equivalence of star products and discuss some of its properties. We will show that it implies equivalence in the case of twist star products. As a consequence of this and of a result by S. Waldmann and H. Bursztyn, we will argue that non-trivial equivariant line bundles and twist star products cannot coexist on the same symplectic manifold. Therefore, one can deduce for example that there is no symplectic star product on the projective space CP^(n-1) induced by a twist based on U(gl_n(C))[[h]]. - Thomas Willwacher: Rational Homotopy Theory of the little disks operads

The little n-disks operads are classical objects in topology, introduced by Boardman-Vogt and May in the 1970's in their study of iterated loop spaces. They have since seen a wealth of applications in algebra and topology, and received much attention recently due to their appearance in the manifold calculus of Goodwillie and Weiss, and relatedly in the factorization (or topological chiral) homology by Lurie, Francis, Beilinson-Drinfeld and others. I report on joint work with Victor Turchin and Benoit Fresse, in which we (mostly) settle the rational homotopy theory of the little n-disks operads, by showing that they are intrinsically formal for n>=3, and by computing the rational homotopy type of the function spaces between these objects. As an application we obtain complete rational invariants of long knots in codimension >=3. - Marco Zambon: Deformations of presymplectic forms

We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an L-infinty algebra, which is a cousin of the Koszul dgLA associated to a Poisson manifold. Its geometric understanding relies on ideas from Dirac geometry. In addition, we show there is a strict morphism from this L-infinty algebra to the one that controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed. This talk is based on joint work with Florian Schätz.