University of Bologna
Title: Convex orderings and quantum tangent spaces.
Abstract: One of the most remarkable phenomena that occurs when we take in consideration a quantum enveloping algebra $U_q(\mathfrak{g})$ is the fact that a PBW basis is not uniquely defined but rather depends upon the choice of a reduced decomposition of the longest element $w_0$ in the corresponding Weyl group. For each of such choices we have a set of so-called quantum root vectors as defined by Lusztig, together with a convex order defined on the positive roots of $\mathfrak{g}$. In this work in collaboration with P. Papi, we study how the combinatorics of a reduced decomposition of $w_0$ is connected to the existence of a quantum tangent space for the full quantum flag manifold of $\mathfrak{g}$.
Newcastle University
Title: Towards bases for representations of QSP coideal subalgebras
Abstract: I will discuss an ongoing project on representations of certain quantum symmetric pair coideal subalgebras of quantum groups. By recent work of Stefan Kolb and Jake Stephens, such algebras have equivalents of roots, satisfy the PBW theorem, and their irreducible finite dimensional representations have weights and can be classified in terms of weights. We would now like to construct bases for these representations compatible with their relationship with quantum groups.
I will explain what are these algebras, why they are of interest, how they relate to quantum groups, and what Kolb and Stephens can show about their representations. I will then describe desired properties of the bases of these representations that one could hope for in analogy with classical Lie theory, list questions about structures which control these bases, and partially answer some of these questions. Joint work in progress with Stefan Kolb.
University of Bologna
Title: Reduction of Quantum Principal Bundles
Abstract: We develop the theory of reduction of quantum principal bundles over projective bases. We show how the sheaf theoretic approach can be effectively applied to certain relevant examples as the Klein model for the projective spaces; in particular we study in the algebraic setting.
Dresden University of Technology
Title: The ring of differential operators on a monomial curve is a Hopf algebroid
Abstract: The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be cocommutative and cocomplete left Hopf algebroid, which essentially means that the category of D-modules is closed monoidal. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode), which means that the subcategory of those D-modules that are finite rank vector bundles over the curve is rigid. Based on joint work with Myriam Mahaman.
University of Vienna
Title: Newton polytopes, quantum affine algebras, and scattering amplitudes
Abstract: In this talk, I will talk about a connection between Newton polytopes, representations of quantum affine algebras, and scattering amplitudes in physics. We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using Newton polytopes. We give a general formula of u-variables which appears in the study of scattering amplitudes in physics using prime tableaux (corresponding to prime modules of quantum affine algebras of type A) and Auslander-Reiten quivers of Grassmannian cluster categories. This is joint work with Nick Early.
University of Zagreb
Title: Reflective localizations of bicategories
Abstract: In this talk, we give an overview of the theory of localization of bicategories via bicalculus of fractions. The bicategory of fractions with respect to a class of morphisms was introduced by Pronk in 1996 as a generalization of the Gabriel-Zisman theory of localization. The main goal was to show that étendues and stacks arise as bicategories of fractions of appropriate categories of groupoids, e.g. orbifolds as localizations of Lie groupoids. We expand on the theory of localization of bicategories by giving a characterization of reflective localizations.
Charles University in Prague
Title: Noncommutative geometry of the quantum flag manifolds: the rank 2 case
Abstract: The quantum flag manifolds are a fascinating class of quantum homogeneous spaces that q-deform the function algebra of the complex flag manifolds. The quantum flag manifolds of irreducible type form one of the best understood classes of noncommutative geometries in the whole theory of quantum groups. This is due in large part to the discovery by Heckenberger and Kolb in the early 2000s that the classical de Rham complex of the irreducible flags admits an essentially unique q-deformation. The question of how to extend beyond the irreducible setting has been the subject of much interest recently. In this talk I will present the state of the art for those quantum flag manifolds associated with simple Lie algebras of rank 2. Moreover, time permitting, I will briefly discuss the rank 3 and 4 cases.
Institute of Mathematics, Czech Academy of Sciences
Title: On existence of coideal deformations of the Kronecker embedding
Abstract: The Kronecker embedding of gl(n)+gl(m) into gl(nm) is defined as differential of the tensor product embedding of GL(n) x GL(m) into GL(nm). We investigate the question of whether it is possible to deform U(gl(n)+gl(m)) into a quantum subgroup of Uq(gl(nm)) with certain structural properties. We draw upon the theory of $\iota$-quantum groups represented by quantum symmetric spaces and primarily look for deformations that are coideal and/or have (quasi-)K-matrix, PBW bases, canonical and crystal bases. The main motivation is a construction of suitable crystal bases for the Kronecker quantum subgroup with application to branching problem for the Kronecker embedding.
University of Zagreb
Title: Dirac operators in representation theory
Abstract: This talk gives an overview of the use of the Dirac operators in representation theory. Dirac operators were introduced into representation theory by Parthasarathy as a tool for constructing discrete series representations. D. Vogan studied an algebraic version of the Dirac operator and in 1997 conjectured that the Dirac cohomology of an admissible $(\mathfrak{g}, K)$--module, if nonzero, determines its infinitesimal character. Vogan's conjecture was proved by Huang and Pandžić in 2002.
In this talk, we will explain the advantages of the use of the Dirac operators in representation theory, and give an overview of some known results with an emphasis on a recent paper by Pandžić, Prlić, Souček, and Tuček which provides an alternative proof of the classification of unitary highest weight modules originally proved by Enright, Howe and Wallach and independently by Jakobsen.
University of Zadar
Title: Global Gauss decomposition of quantum groups
Abstract: Gauss elimination procedure has been extended to linear systems over noncommutative rings. For example, Oystein Ore considered the case of Ore domains in 1931. In a modern expression of the Gauss procedure, Gelfand and Retakh decomposed the generic square matrices with noncommutative entries into the products of lower triangular unidiagonal, diagonal and upper triangular unidiagonal matrices in terms of quasideterminants which are certain well-behaved noncommutative rational functions attached to matrices with noncommutative entries. Quasideterminants involve inverses, hence the decomposition holds only after localizing the algebra. If the matrix is multiplied by a permutation matrix, the new matrix can be decomposed in another localization and, informally speaking, all permutation matrices together determine a global cover of the noncommutative spectrum of the algebra generated by entries of generic matrices. In FRT approach, coordinate rings of quantum groups are given in terms of matrices of generators. For quantum linear groups, the cover by localizations stemming from the noncommutative Gauss decomposition can be fully formalized and the formulas for the entries of the lower triangular matrices in the decomposition extend to algebra maps from lower quantum Borel subgroup to the localizations. Moreover these algebra maps are compatible with induced comodule structure over Borel and become cleavages for Hopf algebraic smash products or, in a geometrical language, local sections of a locally trivial principal bundle over the quantum flag variety. The result appropriately extends to parabolic subgroups inducing a fine tower of bundles among partial quantum flag varieties, and conjecturally it extends to other quantum groups of FRT type. The fully noncommutative case is related to the universal noncommutative flag varieties where the exactness properties of involved localizations hold only for finitely generated projectives, beyond which one needs higher categorical treatment involving Bousfield localization and derived descent.
Charles University in Prague
Title: Realization of generalized Verma modules for quantum groups
Abstract: We introduce suitable technical environment, allowing a (rather concrete and explicit) realization of generalized Verma modules for quantum groups. This is a first step in the analysis of induced representations for quantum groups.