Köln Algebra and Representation Theory Seminar

Abstract: After recalling the definition and some examples of elliptic root systems, I will explain motivations, known facts and some problems on the subject. 

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The Lie algebra gl(∞) is the Lie algebra of infinite matrices with only finitely many nonzero entries; it can also be realised as the direct limit of a sequence of finite dimensional reductive Lie algebras in several different ways.

The algebra gl(∞) is both like and unlike its finite dimensional counterpart. It has Cartan and Borel subalgebras, but their behaviour is quite different than in the finite dimensional case. In particular Borels are not conjugate to each other, and this means that a choice of Borel makes a big difference when studying highest weight representations. I will review some general results on the theory of highest weight representations of gl(∞) from the last few years, up to and including work in progress with Ivan Penkov. 

Abstract: If G is a reductive (connected, complex) Lie group with Lie algebra g, then Hotta-Kashiwara introduced a certain D-module HC on g called the Harish-Chandra D-module. This is precisely the module whose distributional solutions are the invariant eigen-distributions appearing in character theory for real reductive Lie groups. A key result by Hotta-Kashiwara is that HC is semi-simple with simple summands in bijection with the irreducible representations of the Weyl group (the algebraic Springer correspondence). In this talk I'll describe a natural generalization of HC to a D-module on the tangent space of a symmetric space. I'll explain the extent to which Hotta-Kashiwara's results generalise to this setting. This is based on joint work with Levasseur, Nevins and Stafford; I'll assume that the audience doesn't know anything about D-modules.

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Abstract: The theory of Hall algebras has known many spectacular developments and applications since the discovery by Ringel of their connection with quantum groups. One important object arising naturally in the study of Hall algebras is the integration map defined by Reineke, which allows to produce certain celebrated wall-crossing identities. In this talk I will first focus on the Dynkin case and show how the integration map can be interpreted in a natural way via the representation theory of quantum affine algebras. I will then explain how this open perspectives towards an analogous interpretation for more general quivers, relying on the framework of quantum cluster algebras. This is an ongoing joint work with Lang Mou. 

Abstract: Parametrizations of the  canonical bases, string basis and theta basis, can be obtained by the tropicalization of  the Berenstein-Kazhdan decoration function and the Gross-Hacking-Keel-Kontsevich potential respectively. For  a classical Lie algebra and a reduced decomposition $\mathbf i$,  the decorated graphs are constructed algorithmically, vertices of such graphs are labeled by monomials which constitute the set of monomials of the Berenstein-Kazhdan potential.  Due to this algorithm we also get a characterization of $\mathbf i$-trails. The algorithm uses multiplication and summations only, its complexity  is linear in time of writing the monomials of the potential. For type A, there is an algorithm due to Gleizer and Postnikov which gets all monomials of the Berenstein-Kazhdan potential. For type A, our algorithm uses simpler combinatorics and is faster than the Gleizer-Postnikov algorithm. The cluster algorithm due to Genz, Schumann and me is polynomial in time but it uses divisions of polynomials. If time permits I will report on applications of decorated graphs to analysis of the Newton polytopes of F-polynomials related to  the Gross-Hacking-Keel-Kontsevich potentials. This talk is based on joint works with Volker Genz and Bea Schumann and with Yuki Kanakubo and Toshiki Nakashima.