Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics
Herbert Abels and Ernest Vinberg
The workshop is part of the conference program of the DFG-funded CRC 701 "Spectral Structures and Topological Methods in Mathematics" at the University of Bielefeld.
The schedule of the workshop can be found here in pdf-format. (as of July 16th)
(Preliminary list)
| Herbert Abels (Bielefeld University) | Less than two generators suffice In control theory the problem arose if certain Lie groups can be generated as semigroups by two one-parameter subsemigroups. We show in joint work with E. B. Vinberg that actually two one-dimensional compact subgroups suffice. A consequence of our more precise result is that given a non-central elemenet $g$ in a simple Lie group there is an elliptic element $h$ in $G$ such that $h$ and $ghg^{-1}$ generate a dense subsemigroup of $G$. |
| Ivan Arzhantsev (Moscow State University/Ludwig-Maximilians-Universität München) |
The automorphism group of a variety with torus action of complexity one
In 1970, Demazure gave a combinatorial description of the automorphism
group Aut(X) of a complete smooth toric variety X as a linear algebraic
group. The central concept is a root system associated with a complete
fan. Later Cox interprated and generatlized these results in terms of
the homogeneous coordinate ring R(X). We describe the automoprhism group of a complete rational variety X with torus action of complexity one. Our description is based on a presentation of the Cox ring R(X) in terms of trinomials and on an interpretation of Demazure's roots as homogeneous locally nilpotent derivations of R(X). Also we obtain an explicit description of the root system of the semisimple part of Aut(X). The results are applied to the study of almost homogeneous varieties. This is a joint work with Jurgen Hausen, Elaine Herppich and Alvaro Liendo. |
| Roman Avdeev (Moscow State University) | How to classify solvable spherical subgroups in semisimple algebraic groups Solvable spherical subgroups in semisimple algebraic groups were first classified by Luna in 1993. A completely different classification of these subgroups was obtained by the speaker in 2011. Besides, there is a general approach for classifying arbitrary spherical subgroups, which was suggested by Luna in 2001 and has led to a complete classification after decade-long efforts of several researchers. The aim of this talk is to give a sketchy presentation of all the three classifications and interrelations between them. |
| Stephanie Cupit-Foutou (Bochum) | Canonical real structures on wonderful varieties The talk deals with a joint work with D. Akhiezer. The existence and uniqueness of a canonical real structure for homogeneous spherical varieties $G/H$ with $H$ self-normalizing and for their wonderful embeddings will be discussed. |
| Alexander Elashvili (Razmadze Mathematical Institute, Tbilisi) | Cyclic elements in semisimple Lie algebras Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field of characteristic $0$, and let $e$ be a non-zero nilpotent element of $\mathfrak{g}$. By the Morozov-Jacobson theorem the element $e$ can be included in $\mathfrak{sl}_2$-triple $\mathfrak{s}=\{e,h,f\}$. Then the eigenspace decomposition of $\mathfrak{g}$ with respect to $ad(h)$ is a $\mathbb{Z}$-grading of $\mathfrak{g}$. Let $d$ be maximal integer such that $\mathfrak{g}_d:=\{x\in\mathfrak{g} :[h,x]=dx\}$ is not zero. An element of $\mathfrak{g}$ of the form $e+F$, where $F$ is a non-zero element of $\mathfrak{g}_{-d}$ is called a cyclic element, associated with $e$. For an application in the theory of integrable systems it is important to find all nilpotent elements $e$ in semisimple Lie algebras, such that there exists a semisimple cyclic element, associated with $e$. In my talk I'll present a solution of this problem. All results of this talk are obtained jointly with V.G.Kac and E.B.Vinberg. |
| Heinz Helling (Bielefeld) | The Markov group and some class of hyperbolic $3$-manifolds We introduce an associative algebra over the field of complex numbers equipped with the Markov form $xyz - x^2 - y^2 - z^2$. We call this structure "Markov algebra". We exhibit a group of units in the Markov algebra which is isomorphic to the modular group $PSL_2(\mathbb{Z})$. This statement will be proven when visualizing what we call the real Markov surface. We shall describe two applications of the unit group in the Markov algebra. First we show how to construct large degree algebraic number fields with interesting features like class number phenomina. The second concerns hyperbolic manifolds of dimension $3$. We shall describe a recipee which allows to construct systematically trace fields. |
| Ruth Kellerhals (Fribourg) | Hyperbolic Coxeter groups and minimal growth rate After a short introduction to hyperbolic Coxeter groups of finite covolume and their growth series, we present new results about Coxeter groups acting on hyperbolic 3-space with minimal growth rate. Of special interest are the Coxeter orthoschemes [3,5,3] and [3,3,6]. Part of this work is joint with A. Kolpakov. |
| Alexander Kolpakov (Fribourg) | On the optimality of the ideal right-angled 24-cell We will show that among all four-dimensional ideal right-angled hyperbolic polytopes the 24-cell is of minimal volume and of minimal facet number. |
| Dmitri Panyushev (Moscow) | Commuting involutions and degenerations of isotropy representations Let $\sigma_1$ and $\sigma_2$ be commuting involutions of a semisimple algebraic group $G$. This yields a $\mathbb{Z}_2\times \mathbb{Z}_2$-grading of $\mathfrak{g}=\operatorname{Lie}(G)$, $\mathfrak{g}=\bigoplus_{i,j=0,1}\mathfrak{g}_{ij}$, and we study invariant-theoretic aspects of this decomposition. Let $\mathfrak{g}\langle\sigma_1\rangle$ be the $\mathbb{Z}_2$-contraction of $\mathfrak{g}$ determined by $\sigma_1$. Then both $\sigma_2$ and $\sigma_3:=\sigma_1\sigma_2$ remain involutions of the non-reductive Lie algebra $\mathfrak{g}\langle\sigma_1\rangle$. The isotropy representations related to $(\mathfrak{g}\langle\sigma_1\rangle, \sigma_2)$ and $(\mathfrak{g}\langle\sigma_1\rangle, \sigma_3)$ are degenerations of the isotropy representations related to $(\mathfrak{g}, {\sigma_2})$ and $(\mathfrak{g}, {\sigma_3})$, respectively. These degenerated isotropy representations retain many good properties. For instance, they always have a generic stabiliser and their algebras of invariants are often polynomial. We also develop some theory on Cartan subspaces for various $\mathbb{Z}_2$-gradings associated with the $\mathbb{Z}_2\times \mathbb{Z}_2$-grading of $\mathfrak{g}$ and study the special case in which $\sigma_1$ and $\sigma_2$ are conjugate. |
| Gregory Soifer (Bar-Ilan University, Tel Aviv) |
Embedding $\mathbb{Z}^2 \ast \mathbb{Z}$ as a discrete subgroup into small linear groups
At the 2011 Durham Conference " Geometry and Arithmetic of Lattices M. Kapovich formulated the following questions Question 1 Let $\Gamma =SL(3, \mathbb{Z})$. What are the torsion free finitely generated subgroups of $\Gamma$? Question 2 Does there exist an embedding $\mathbb{Z}^2\ast\mathbb{Z} \hookrightarrow SL(3, \mathbb{Z})$? Recall that in our joint paper with G.Margulis "Maximal subgroups of infinite index in finitely generated linear group", J. Algebra 69, No 1., (1981), we proved in particular that for all $m \in \mathbb{Z}, m > 0$, there exists an embedding $\mathbb{Z}^2 \ast F_m\hookrightarrow SL(n, \mathbb{Z}) , n \geq 4, $ where $F_m$ is the free group of rank $m$. The main goal of the talk is to explain the main ideas and methods we used in the proof of the following Main Theorem If $p $ and $m$ are arbitrary positive integers then there exists an embedding $\mathbb{Z}^2\ast F_m\hookrightarrow SL(3, \mathbb{Z}[1/p])$. |
| Dmitry Timashev (Moscow State Univrsity)) | On quotients of affine spherical varieties by unipotent subgroups Let $G$ be a connected complex reductive algebraic group and $X$ be an affine spherical $G$-variety. Then any Borel subgroup $B\subset G$ acts on $X$ with finitely many orbits. Now suppose that $H\subset B$ is a normal unipotent subgroup such that the algebra $\mathbb{C}[G/H]$ is finitely generated. Then $\mathbb{C}[X]^H$ is finitely generated, too, and one may consider the categorical quotient map $\pi:X\to X/H:=\mathrm{Spec}\;\mathbb{C}[X]^H$. Clearly, $B/H$ acts on $X/H$. Recently Panyushev posed a question whether the number of orbits for this action is always finite. (The point is that $\pi$ is not always surjective.) He conjectured that this is always the case for $H=[U,U]$, the commutator of the maximal unipotent subgroup $U\subset B$. We discuss an approach to answering the question, give a positive answer in some cases, and provide a counterexample to the above conjecture. |
| Ernest Vinberg (Moscow State University) | Affine hyperbolic reflection groups. Let $\mathbb{R}^{n-1,1}$ be the Minkowski space, a real vector space with a scalar product of signature $(n-1,1)$, and let $\mathbb{C}^{n-1,1}$ be its complexification. Let $C$ be the future cone in $\mathbb{R}^{n-1,1}$. The "Siegel domain" $S_n=\{z=x+iy\in C^{n-1,1}| y\in C\}$ is a model of the Hermitian symmetric space $D_n=O'(n,2)/(O(n)\times SO(2))$, a symmetric domain of type IV. More precisely, it is associated to an isotropic vector $v$ of the space $\mathbb{R}^{n,2}$, where the group $O(n,2)$ acts. The stabilzer of v is the semidirect product of $\mathbb{R}^{n-1,1}$ and $O'(n-1,1)$ (the index $2$ subgroup of $O(n-1,1)$ leaving invariant the future cone $C$) and acts in $S_n$ by affine transformations. Let $\Gamma\subset O'(n,2)$ be an arithmetic lattice. Then the Baily-Borel compactification $\overline{D_n/\Gamma}$ is the projective spectrum of the (graded) algebra $A(D_n,\Gamma)$ of automorphic forms. By analogy with finite linear groups, it would be of great interest to find all the groups $\Gamma$ with free algebras of automorphic forms. Leaving aside the classical cases $n=1,2$, only one such example due to Igusa (1962), with $n=3$, was known until recently. In 2010, the speaker proved freeness of the algebras of automorphic forms for some groups $\Gamma$ in dimensions $n=4,5,6,7$. If the algebra $A(D_n,\Gamma)$ is free, then $\overline{D_n/\Gamma}$ is a weighted projective space and is non-singular at any "cusp" corresponding to a rational isotropic vector $v\in \mathbb{R}^{n,2}$. The local structure of $\overline{D_n/\Gamma}$ at such cusp is controlled by the stabilizer $\Gamma_v$ of $v$ in $\Gamma$ (the "cusp group"), which is a discrete group of affine transformations of the Siegel domain $S_n$. The intersection of $\Gamma_v$ with $\mathbb{R}^{n-1,1}$ is a lattice in $\mathbb{R}^{n-1,1}$, and the projection $\Delta_v$ of $\Gamma_v$ to $O'(n-1,1)$ is an arithmetic lattice in $O'(n-1,1)$ acting in the Lobachevsky space $L^{n-1}$ as a discrete group with a fundamental domain of finte volume. The local ring $O_v$ of $\overline{D_n/\Gamma}$ at $v$ can be desribed in terms of the cusp group. It can be regular, only if the group $\Gamma_v$, and hence the group $\Delta_v$, is generated by reflections. Due to results of the speaker and F. Esselmann, this implies that $n\leq 22$ (assuming that there are cusps, i.e. the lattice $\Gamma$ is not cocompact). However, if even the cusp group is generated by reflections, the ring $O_v$ need not be regular. In 1981, O.V. Shvartsman proved that if the group $\Gamma_v$ is generated by reflections and the fundamental domain of the group $\Delta_v$ in $L^{n-1}$ is a simplex, then the ring $O_v$ is regular. He conjectured that this condition is also necessary. This is still an open question. The positive answer would give a better estimate $n\leq 10$. |
| Jörg Winkelmann (Ruhr-Universität Bochum) | Holomorphic curves and integral points on principal bundles The theorems of Faltings, Siegel and the uniformization theorem for Riemann surfaces show that for an algebraic variety of dimension one defined over a number field K the following conditions are equivalent: (1) Every holomorphic map from the complex line into the associated complex space is constant. (2) The set of all integral points (= points with coordinates in the ring of algebraic integers of the respective number fields) on the variety is finite for every finite field extension of K. It is conjectured by Lang and Vojta that similar equivalences hold in higher dimensions. Wee investigate this conjecture in the context of algebraic groups and namely verify such an equivalence for certain principal bundles of algebraic groups. |
| Oksana Yakimova (Jena University) | Symmetric invariants of semi-direct products (four non-surjective cases) Let $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ be a symmetric decomposition of a simple Lie algebra $\mathfrak{g}$. We can contract $\mathfrak{g}$ to a semi-direct product $\mathfrak{s}=\mathfrak{g}_0\ltimes \mathfrak{g}_1$, where $\mathfrak{g}_1$ becomes an Abelian ideal. D.I. Panyushev has conjectured that this new (non-reductive) Lie algebra still has a free algebra of symmetric invariants generated by the same number, $rk \mathfrak{g}$, of homogeneous polynomials. The conjecture is known to be true, when the restriction map $C[\mathfrak{g}]^G \to C[\mathfrak{g}_1]^{G_0}$ is surjective. In that case generators for $\mathfrak{s}$ can be obtained from the generators of $S(\mathfrak{g})^\mathfrak{g}$. There are four "non-surjective" pairs and to get an answer for them one needs a different technique. |
| Vladimir Zhgoon (Institute of System Studies, Moscow) |
On generation of the little Weyl group by reflections and products of orthogonal reflections
(based on a joint survey with D.A.Timashev) Let $G$ be a reductive group over an algebraically closed field of characteristic zero, $B$ be a Borel subgroup of $G$, and $X$ be a normal $G$-variety. We consider the so-called extended little Weyl group of $X$, which is an important invariant introduced by F. Knop. This group is a semidirect product of the little Weyl group and the Weyl group of the Levi part of the normalizer of a general $B$-orbit. The extended little Weyl group can be embedded in the Weyl group $W$ of $G$. We show that the extended little Weyl group as a subgroup of $W$ is generated by reflections and products of two orthogonal reflections. This generalizes a result of Brion for spherical varieties and gives another proof of the fact that the little Weyl group is generated by reflections. |
For registration please use our online registration form.
All the talks take place in (V2-210).
A limited number of rooms has been reserved at the university apartments on campus.The other option is the hotel Arcadia in town. If you want to make a reservation please contact Ms. Anita Cole cole(at)math(dot)uni-bielefeld(dot)de or Ms. Britta Heidrich heidrich(at)math(dot)uni-bielefeld(dot)de
There will be a conference dinner on Monday July 23 at 18:30 in IBZ (on campus). When registering please indicate if you will participate in the dinner.
Please contact the organizers, Mr. Abels abels(at)math(dot)uni-bielefeld(dot)de or Mr. Vinberg vinberg(at)math(dot)uni-bielefeld(dot)de, for further information.