Lie Groups and Algebraic Groups - uni-bielefeld.de · Lie Groups and Algebraic Groups 22–24July2015 Faculty of Mathematics Bielefeld University Lecture Room: V3-201...

Lie Groups and Algebraic Groups

22–24 July 2015

Faculty of MathematicsBielefeld University

Lecture Room: V3-201

This workshop is part of the DFG-funded CRC 701Spectral Structures and Topological Methods in Mathematics

at Bielefeld University

Organisers: Herbert Abels and Ernest Vinberg

www.math.uni-bielefeld.de/sfb701/2015_LieGroups/

Lie Groups and Algebraic Groups

Schedule: Wednesday July 22nd

Lecture Room: V3-201

10:30–11:30 Mark Sapir (Vanderbilt)On groups with Rapid Decay

11:30–12:00 Coffee Break (Room V3-201)

12:00–13:00 Hannah Bergner (Bochum)Actions on supermanifolds and automorphism groups of compact complexsupermanifolds

13:00–15:30 Lunch Break

15:30–16:30 Pavel Tumarkin (Durham)Reflection groups via cluster algebras

16:30–17:00 Coffee Break (Room V3-201)

17:00–18:00 Anna Felikson (Durham)Cluster algebras via reflection groups

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Lie Groups and Algebraic Groups

Schedule: Thursday July 23rd

Lecture Room: V3-201

10:00–11:00 Dmitri Panyushev (Moscow)Minimal inversion complete sets and maximal abelian ideals

11:00–11:30 Coffee Break (Room V3-201)

11:30–12:30 Alexander Elashvili (Tbilisi)On Lieandric numbers

12:30–15:00 Lunch Break

15:00–16:00 Matthieu Jacquemet (Fribourg)Hyperbolic truncated simplices and reflection groups

16:00–16:30 Coffee Break (Room V3-201)

16:30–17:30 Christian Lange (Cologne)Towards a generalized fixed point theorem

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Lie Groups and Algebraic Groups

Schedule: Friday July 24th

Lecture Room: V3-201

10:00–11:00 Willem de Graaf (Trento)Real and complex nilpotent orbits of so(4,4)

11:00–11:30 Coffee Break (Room V3-201)

11:30–12:30 Ernest Vinberg (Moscow)“Good” reflection groups in O(2,n)

12:30–15:00 Lunch Break

15:00–16:00 Valdemar Tsanov (Göttingen)Momentum images of representations, secant varieties and invariants

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Lie Groups and Algebraic Groups

Abstracts

Hannah Bergner (Bochum)Actions on supermanifolds and automorphism groups of compact complex supermanifoldsLetM be a supermanifold and let g be a finite-dimensional Lie subsuperalgebra of the Liesuperalgebra Vec(M) of super vector fields onM. We study the question in which cases sucha Lie superalgebra g of super vector fields onM is induced by an action of a Lie supergroup.Necessary and sufficient conditions for this are provided, generalizing the results of Palais in theclassical case.

In the special case of a compact complex supermanifold M, the Lie superalgebra Vec(M)of super vector fields onM is finite-dimensional. By a result of Bochner and Montgomery theautomorphism group of a compact complex manifold M carries the structure of a complex Liegroup whose Lie algebra is isomorphic to the Lie algebra of vector fields on M. We investigatehow the automorphism group of a compact complex supermanifoldM can be defined and provethat it carries the structure of a complex Lie supergroup with Lie superalgebra Vec(M). (Partof this work is joint with M. Kalus.)

Alexander Elashvili (Tbilisi)On Lieandric numbersLieandric numbers count the number of biparabolic Lie subalgebras of index 1 in full matrixalgebras (these are examples of Frobenius Lie algebras, providing constant solutions of theYang-Baxter equations). In the talk, a combinatoric description of these numbers will be given, aconjecture concerning their asymptotics will be formulated and some evidence for the conjecturewill be presented. This is a joint work with M. Jibladze.

Anna Felikson (Durham)Cluster algebras via reflection groupsCluster algebras are defined inductively via repeatedly applied operation of mutation. Duringthe last decade it turned out that the formula of mutation appears in various contexts. We willuse linear reflection groups to build a geometric model for acyclic cluster algebras, where “partial”reflections will play the role of mutations.

Willem de Graaf (Trento)Real and complex nilpotent orbits of so(4,4)

Let G(k) denote the direct product of four copies of SL(2,k). This group acts on the fourthtensor power of k2. We consider the nilpotent orbits of this action when k is the complex andthe real field. We briefly indicate the physical relevance of these orbits. Then we discuss methodsto list them. These are based on the fact that the given representation of G(k) can be realizedas a theta-representation in the simple complex Lie algebra of type D4. It is well known thatin the complex case there are 30 nilpotent orbits. It turns out that when k=R there are 145nilpotent orbits (excluding 0).

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Lie Groups and Algebraic Groups

Matthieu Jacquemet (Fribourg)Hyperbolic truncated simplices and reflection groupsA hyperbolic truncated simplex is obtained as polarly truncated finite-volume part of a so-calledtotal simplex in the extended hyperbolic space. The class of hyperbolic truncated simplicescontains interesting polytopes, such as the hyperbolic Coxeter pyramid P17⊂H17 related tothe orientable hyperbolic arithmetic orbifold of absolute minimal volume. In this talk, we shalldiscuss geometric, algebraic and arithmetic features of Coxeter hyperbolic truncated simplices.

Christian Lange (Cologne)Towards a generalized fixed point theoremIt is well known that isotropy groups of finite real reflection groups are generated by the reflectionsthey contain. The same is true for isotropy groups of unitary reflection groups due to Steinberg’sfixed-point theorem. In the talk we explain that these results are special cases of a more generalfixed point theorem, whose proof, however, still relies on a classification and on cumbersomecomputations. We sketch what is known and discuss illustrating examples.

Dmitri Panyushev (Moscow)Minimal inversion complete sets and maximal abelian idealsLet g be a simple Lie algebra, b a fixed Borel subalgebra, and W the Weyl group of g. I amgoing to speak about a relationship between the maximal abelian ideals of b and the minimalinversion complete sets of W . The latter have been recently introduced by Malvenuto, MosenederFrajria, Orsina, and Papi.

Mark Sapir (Vanderbilt)On groups with Rapid DecayThe property of Rapid Decay (RD) of groups and group actions was introduced by Haagerup and isvery important for analytic application of groups. The property is also very geometric and so it is ofinterest to geometric group theorists. I will survey results and methods in this area of group theory.

Valdemar Tsanov (Göttingen)Momentum images of representations, secant varieties and invariantsWe address the following question: What is the momentum image of an irreducible unitaryrepresentation of a compact Lie group? Despite the extensive general theories on momentaand representations, where the given case takes a central place, there lacks, to the best of ourknowledge, a computable explicit description of the momentum image in terms of the invariantsof the representation, say, the highest weight. Many cases are known, but many important onesstill escape. I will discuss an approach based on works of Wildberger, Sjamaar and Heinzner.I will also discuss relations to secant varieties of embedded flag varieties and degrees of invariantpolynomials. This is joint work with E. Hristova and T. Maciazek.

Pavel Tumarkin (Durham)Reflection groups via cluster algebrasI will describe a construction of presentations of finite and affine Weyl groups arising from clusteralgebras. In particular, this leads to presentations of Weyl groups as quotients of various Coxetergroups. I will also discuss some generalizations.

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Lie Groups and Algebraic Groups

Ernest Vinberg (Moscow)“Good” reflection groups in O(2,n)

According to a classical result of Shephard–Todd–Chevalley, finite linear complex reflection groupsare characterized by the property that their algebra of polynomial invariants is free. If we considerthese groups as acting on the corresponding projective spaces (which are Hermitian symmetricspaces of positive curvature), a natural infinite analogue of them are cofinite discrete reflectiongroups in symmetric domains (Hermitian symmetric spaces of negative curvature), the analogueof polynomials being automorphic forms.

The only symmetric domains admitting reflections are complex balls B(n)=U(1,n)/(U(1)×U(n))(of rank 1) and domains of Cartan type IV D(n)=O+(2,n)/(SO(2)×O(n)) (of rank 2). Manyexamples of cofinite discrete reflection groups in B(n) and D(n) for small n are known. Forsome of them it is known that the algebra of automorphic forms is free. However, due to ageneral result of Margulis for symmetric spaces of rank >1, any cofinite discrete group in D(n)containing a reflection, has a finite index subgroup generated by reflections. This means thatthere are a lot of cofinite discrete reflection groups in D(n) for any n, and it is not likelyhoodthat all of them share any good properties. To distinguish “good” reflection groups Γ⊂O(2,n),one can require that dimH2(Γ,Q)=1. Under this condition, there exists a semi-automorphicform (possibly, of fractional weight) vanishing (with multiplicity 1) exactly at the mirrors ofreflections contained in Γ (an analogue of the Vandermonde determinant). Hopefully, this willpermit to prove that, for such “good” reflection groups in O(2,n), the algebra of automorphicforms is free.

In particular, letOd is the ring of integers of the quadratic fieldQ(√d), and σ be its involution. The

extended Hilbert modular group Γd=〈PSL(2,Od),σ〉 is a cofinite discrete group in the domainD(2)(which is the direct product of two copies of the hyperbolic plane). It is often generated by reflections.One can try to calculateH2(Γd,Q), making use of a presentation of Γd obtained in a geometric way.This program was realised for d=2 with the result that the group Γ2 is “good” in the above sense.

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Lie Groups and Algebraic Groups

Participant List

Herbert Abels (Bielefeld)Hannah Bergner (Bochum)Stephanie Cupit-Foutou (Bochum)Alexander Elashvili (Tbilisi)Anna Felikson (Durham, Great Britain)Willem de Graaf (Trento, Italy)Werner Hoffmann (Bielefeld)Matthieu Jacquemet (Fribourg, Switzerland)Matthias Kalus (Bochum)Christian Lange (Cologne)Dmitri Panyushev (Moscow)Detlev Poguntke (Bielefeld)Mark Sapir (Vanderbilt University, USA)Gregory Soifer (Bar-Ilan, Israel)Valdemar Tsanov (Bochum)Pavel Tumarkin (Durham, Great Britain)Ernest Vinberg (Moscow)

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