Abstracts

Talks

• Paolo Aschieri: Atiyah sequences of braided Lie algebras and their splittings
  
  Associated with an equivariant noncommutative principal bundle we give an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. On the principal bundle of orthonormal frames over the quantum sphere S²ⁿ_θ, the splitting of the sequence leads to a Levi-Civita connection on the corresponding module of braided derivations. We present its Riemannian geometry. Joint work with C. Pagani and G. Landi.
• Alessandro Carotenuto: Convex orderings on quantum root vectors and differential calculi
  
  In their recent work Ó Buachalla and Somberg gave a construction of a quantum holomorphic calculus for full quantum flag manifolds of type A. They made use of the celebrated theory of Lusztig automorphisms to prove the existence of a reduced decomposition for the longest element of the Weyl group, such that the generators of the corresponding PBW basis give a quantum tangent space. Preliminary investigations suggest that such reduced decompositions can be found also for type BCD. In this early-stage project, I suggest a way to prove the existence of quantum tangent spaces by looking solely at the combinatorial data of a given reduced decomposition, describing the coproduct formula of quantum root vectors in terms of the convex order induced on the set of positive roots.
• Fredy Diaz Garcia Spectrum of the Dolbeault-Dirac Laplace operator on zero forms for the quantum quadrics
  
  In this talk we report some results about the asymptotic behaviour of the spectrum of the Dolbeault-Dirac Laplace operators associated to the Kähler structure of the irreducible quantum flag manifolds of type B_n or D_n on zero forms by using properties of the differential calculus of Heckenberger and Kolb and representation theory of the quantized enveloping algebras U_q(g).
• Rita Fioresi: Quantum Principal Bundles over Projective Bases
  
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• Niels Kowalzig: Crossed products for universal enveloping algebras of Lie algebroids
  
  We extend a theorem, originally formulated by Blattner-Cohen-Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to universal enveloping algebras of projective Lie-Rinehart algebras: for any given curved (resp. flat) connection, that is, a splitting of a certain kind of a Lie-Rinehart algebra extension, we provide a crossed product decomposition of the associated universal enveloping algebra, and vice versa. As a geometric example, we describe the associative algebra generated by the invariant vector fields on the total space of a principal bundle as a crossed product of the algebra generated by the vertical ones and the algebra of differential operators on the base.
• Andrey Krutov: Cubic Dirac operator for U_q(sl₂)
  
  Alekseev and Meinrenken developed a noncommutative equivariant de Rham theory for homogeneous manifolds equipped with a transitive action of a Lie group G. This led to the definition of a noncommutative Weil algebra related to the Lie algebra g of G. If the Lie algebra g admits a nondegenerate invariant bilinear form, then the noncommutative Weil algebra comes equipped with a cubic Dirac element. Since this Dirac element squares to a scalar, it can be seen as an algebraic Dirac operator. Such Dirac operators have numerous applications in geometry and representation theory. We define the q-deformed noncommutative Weil algebra for U_q(sl₂) and the corresponding cubic Dirac element. We calculate the spectrum of the Dirac operator and the corresponding Dirac cohomologies. We discuss a possible q-defomed generalisation of g-differential algebras.
  This is joint work with Pavle Pandžić (Zagreb)
• Max Kurkov: Lie-Poisson gauge theory
  
  Poisson gauge formalism provides a semiclassical description of noncommutative U(1) gauge theory. We focus on a Poisson gauge theory with a generic Poisson structure of Lie algebra type. We consider the symplectic geometric interpretation of the construction. We discuss the Lagrangian and non-Lagrangian dynamics, and apply our findings to the k-Minkowski case.
• Giuseppe Marmo: Noncommutative probability spaces and differential geometry
  
  Probability operators, the name given by Schwinger to quantum states, possess a very interesting geometrical structure when we consider finite quantum systems. With the finiteness condition the geometry emerges more clearly and in addition we avoid all technicalities due to infinite dimensions. We shall describe some geometrical tensor fields associated with the algebraic properties of random variables and their automorphisms, along with the description of infinitesimal generators of Markovian semigroups.
• Kevin Josué Rodriguez Portillo: Dobeault-Dirac operator on quantum Gr(2,4)
  
  An explicit description of the Dobeault-Dirac operator on quantum Gr(2,4) will be given by using the quantum Bernstein-Gelfand-Gelfand resolution of Heckenberger and Kolb. To determine the spectrum of the Dolbeault-Dirac operator, it is essential to determine the branching laws of SU(4) with respect to the Levi factor. This can be achieved by computing the dimensions of specific weight spaces of the irreducible representations of SU(4). The strategy will be shown in examples as it is still work in progress.
• Réamonn Ó Buachalla: The noncommutative complex geometry of the full flag manifold of quantum SU(3)
  
  We construct a q-deformation of the Dolbeault double complex of the full quantum flag manifold of SU(3). This extends the Heckenberger--Kolb differential calculus of the quantum projective plane, and the recently constructed q-deformed anti-holomorphic complex of the full quantum flag manifold. We present a number of new phenomena which do not occur in the Heckenberger-Kolb setting, such as connections with torsion and non-integrable almost-complex structures.
• Luca Schiavone: The inverse problem for gauge theories via coisotropic embeddings
  
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• Karen R. Strung: Realising Quantum Flag Manifolds as Graph C*-Algebras
  
  In this talk I will show how the C*-completions of the so-called quantum flag manifolds — noncommutative spaces arising as homogeneous spaces of quantum groups — can be realised as graph C*-algebras. After recalling the definition of a quantum flag manifold and its C*-algebra, I will describe how to compute the primitive ideal space using Dijkhuizen and Stokmann's description of a complete set of irreducible *-representations. This allows one to to construct a graph directly from the Weyl group of the associated Lie algebra, and appeal to classification results of Eilers, Ruiz and Sorensen to show that this graph C*-algebra is isomorphic to the C*-algebra of the relevant quantum flag manifold. This recovers some known isomorphisms between the C*-algebras of quantum flag manifolds, as well as determining surprising new ones.
  Joint work with Tomasz Brzeziński, Ulrich Krähmer, and Réamonn Ó Buachalla.
• Alfonso Tortorella: A rigidity result for coisotropic submanifolds in contact geometry
  
  In this talk, based on joint work with Stephane Geudens, we study coisotropic deformations of a compact regular coisotropic submanifold C in a contact manifold (M,H). Our main result states that C is rigid among nearby coisotropic submanifolds whose characteristic foliation is diffeomorphic to that of C. When combined with a classical rigidity result for foliations, this yields conditions under which C is rigid among all nearby coisotropic submanifolds.
• Elmar Wagner: Dobeault-Dirac operator on irreducible quantum flag manifolds via the BGG-resolution and a Parthasarathy-type formula
  
  It will be shown that a quantum generalization of the Bernstein-Gelfand-Gelfand resolution by I. Heckenberger and S. Kolb leads directly to a description of the Dobeault-Dirac operator on irreducible quantum flag manifolds. This description allows for a Parthasarathy-type formula for the Laplacian on 0-forms and on the (0,n)-top forms. As no general result has been established yet, these observations will be discussed using the examples of the quantum Gr(2,4) and the B2-irreducible quantum flag manifold.
• Thomas Weber: On the Durdevic approach to quantum principal bundles
  
  We present part of the noncommutative differential geometry framework developed by M. Durdevic in the 90's. Given a principal comodule algebra (i.e. a faithfully flat Hopf-Galois extension) and a bicovariant differential calculus on the structure Hopf algebra, we construct vertical forms of all degrees and prove that they admit an extension of the coaction as a morphism of differential graded algebras. Choosing a certain "complete" calculus on the total space algebra, we construct horizontal and base forms and discuss the emerging noncommutative Atiyah sequence and graded Hopf-Galois extension. The approach is compared to quantum principal bundles à la Brzeziński-Majid and exemplified for calculi on crossed product algebras.