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Walter Freyn

Publications and source records attributed to Walter Freyn.

13 recordsLinked to original sources
arXiv2019

Survey on real forms of the complex $A_2^{(2)}$-Toda equation and surface theory

The classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for $k$-symmetric spaces over reductive Lie groups. In this survey we will show that to each of the five different types of real forms for a loop group of $A_2^{(2)}$ there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in $\mathbb {CP}^2$, minimal Lagrangian surfaces in $\mathbb {CH}^2$, timelike minimal Lagrangian surfaces in $\mathbb {CH}^2_1$, proper definite affine spheres in $\mathbb R^3$ and proper indefinite affine spheres in $\mathbb R^3$, respectively.

math.DG↗
arXiv2017

Kac-Moody symmetric spaces

In the present article we introduce and study a class of topological reflection spaces that we call Kac-Moody symmetric spaces. These generalize Riemannian symmetric spaces of non-compact type. We observe that in a non-spherical Kac-Moody symmetric space there exist pairs of points that do not lie on a common geodesic; however, any two points can be connected by a chain of geodesic segments. We moreover classify maximal flats in Kac-Moody symmetric spaces and study their intersection patterns, leading to a classification of global and local automorphisms. Unlike Riemannian symmetric spaces, non-spherical non-affine irreducible Kac-Moody symmetric spaces also admit an invariant causal structure. For causal and anti-causal geodesic rays with respect to this structure we find a notion of asymptoticity, which allows us to define a future and past boundary of such Kac-Moody symmetric space. We show that these boundaries carry a natural polyhedral structure and are cellularly isomorphic to the halves of the geometric realization of the twin buildings of the underlying split real Kac-Moody group. We also show that every automorphism of the symmetric space is uniquely determined by the induced cellular automorphism of the future and past boundary. The invariant causal structure on a non-spherical non-affine irreducible Kac-Moody symmetric space gives rise to an invariant pre-order on the underlying space, and thus to a subsemigroup of the Kac-Moody group. We conclude that while in some aspects Kac-Moody symmetric spaces closely resemble Riemannian symmetric spaces, in other aspects they behave similarly to ordered affine hovels, their non-Archimedean cousins.

math.GR↗
arXiv2016

A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group

Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra g = g(A) and (adjoint) Kac-Moody group G = G(A)=$\langle\exp(ad(t e_i)), \exp(ad(t f_i)) \,|\, t\in C\rangle$ where $e_i$ and $f_i$ are the simple root vectors. Let $(B^+, B^-, N)$ be the twin BN-pair naturally associated to G and let $(\mathcal B^+,\mathcal B^-)$ be the corresponding twin building with Weyl group W and natural G-action, which respects the usual W-valued distance and codistance functions. This work connects the twin building of G and the Kac-Moody algebra g in a new geometrical way. The Cartan-Chevalley involution, $\omega$, of g has fixed point real subalgebra, k, the 'compact' (unitary) real form of g, and k contains the compact Cartan t = k $\cap$ h. We show that a real bilinear form $(\cdot,\cdot)$ is Lorentzian with signatures $(1, \infty)$ on k, and $(1, n -1)$ on t. We define $\{x\in {\rm k} \,|\, (x, x) \leq 0\}$ to be the lightcone of k, and similarly for t. Let K be the compact (unitary) real form of G, that is, the fixed point subgroup of the lifting of $\omega$ to G. We construct a K-equivariant embedding of the twin building of G into the lightcone of the compact real form k of g. Our embedding gives a geometric model of part of the twin building, where each half consists of infinitely many copies of a W-tessellated hyperbolic space glued together along hyperplanes of the faces. Locally, at each such face, we find an $SU(2)$-orbit of chambers stabilized by $U(1)$ which is thus parametrized by a Riemann sphere $SU(2)/U(1)\cong S^2$. For n = 2 the twin building is a twin tree. In this case, we construct our embedding explicitly and we describe the action of the real root groups on the fundamental twin apartment. We also construct a spherical twin building at infinity, and construct an embedding of it into the set of rays on the boundary of the lightcone.

math.GR↗
arXiv2014

Weyl group orbits on Kac--Moody root systems

Let $\mathcal{D}$ be a Dynkin diagram and let $\Pi=\{\alpha_1,\dots ,\alpha_{\ell}\}$ be the simple roots of the corresponding Kac--Moody root system. Let $\mathfrak{h}$ denote the Cartan subalgebra, let $W$ denote the Weyl group and let $\Delta$ denote the set of all roots. The action of $W$ on $\mathfrak{h}$, and hence on $\Delta$, is the discretization of the action of the Kac--Moody algebra. Understanding the orbit structure of $W$ on $\Delta$ is crucial for many physical applications. We show that for $i\neq j$, the simple roots $\alpha_i$ and $\alpha_j$ are in the same $W$--orbit if and only if vertices $i$ and $j$ in the Dynkin diagram corresponding to $\alpha_i$ and $\alpha_j$ are connected by a path consisting only of single edges. We introduce the notion of `the Cayley graph $\mathcal{P}$ of the Weyl group action on real roots' whose connected components are in one-to-one correspondence with the disjoint orbits of $W$. For a symmetric hyperbolic generalized Cartan matrix $A$ of rank $\geq 4$ we prove that any 2 real roots of the same length lie in the same $W$--orbit. We show that if the generalized Cartan matrix $A$ contains zeros, then there are simple roots that are stabilized by simple root reflections in $W$, that is, $W$ does not act simply transitively on real roots. We give sufficient conditions in terms of the generalized Cartan matrix $A$ (equivalently ${\mathcal D}$) for $W$ to stabilize a real root. Using symmetry properties of the imaginary light cone in the hyperbolic case, we deduce that the number of $W$--orbits on imaginary roots on a hyperboloid of fixed radius is bounded above by the number of root lattice points on the hyperboloid that intersect the closure of the fundamental region for $W$.

math.GR↗
arXiv2013

Kac-Moody symmetric spaces of Euclidean type

We investigate in detail the class of Euclidean affine Kac-Moody symmetric spaces and their orthogonal symmetric affine Kac-Moody algebras (OSAKAs). These spaces are the only class of Kac-Moody symmetric spaces, that is not directly derived from affine Kac-Moody algebras in the classical sense.

math.DG↗
arXiv2013

Holomorphic completions of affine Kac-Moody groups

We construct holomorphic loop groups and their associated affine Kac-Moody groups and prove that they are tame Fr\'echet manifolds. These results form the functional analytic basis for the theory of affine Kac-Moody symmetric spaces, presented first in the authors thesis. Our approach also solves completely the problem of complexification of loop groups; it allows a complete description of complex Kac-Moody groups and their non-compact real forms.

math.FA↗
arXiv2013

Orthogonal affine Kac-Moody algebras

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues known as Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side: More precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification.

math.DG↗
arXiv2013

Tame Fr\'echet submanifolds

We introduce the new class of submanifolds of co-Banach type in tame Fr\'echet manifolds and construct tame Fr\'echet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fr\'echet manifolds. The results presented are key ingredients in the construction of affine Kac-Moody symmetric spaces; they have also important applications in the study of isoparametric submanifolds in tame Fr\'echet spaces.

math.DG↗
arXiv2011

Affine Kac-Moody symmetric spaces

Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody symmetric spaces are infinite dimensional symmetric spaces associated to affine Kac-Moody groups. They have the structure of tame Fr\'echet manifolds; the natural Ad-invariant scalar product on affine Kac-Moody algebras is Lorentzian, making affine Kac-Moody symmetric spaces into Lorentzian symmetric spaces. Similar to affine Kac-Moody groups sharing most of their structure properties with simple Lie group, also Kac-Moody symmetric spaces share most of their structure properties with their finite dimensional Riemannian counterparts. In particular the classification of Kac-Moody symmetric spaces follows the lines of the classification of finite dimensional Riemannian symmetric spaces: There are four types distinguished, which fall into the two classes of Kac-Moody symmetric spaces of the compact type and of the noncompact type. Symmetric spaces of the compact type and of the noncompact type are related by a duality relation. In addition the geometry of the rank 1-building blocks is the same for Kac-Moody symmetric spaces as for finite dimensional Riemannian symmetric spaces. Kac-Moody symmetric spaces appear in mathematics and theoretical physics: for example their isotropy representations are essentially equivalent to polar actions on Hilbert spaces; twin cities can be embedded equivariantly into the tangent space. In theoretical physics Kac-Moody symmetric spaces got recently a prominent place due to various conjectures relating them to certain formulations of supergravity theories and M-theory.

math.DG↗
arXiv2011

Linear representations of twin cities

For spherical Tits buildings of the classical types there are well-known explicit descriptions as flag complexes. Similarly for affine buildings of the classical types there are explicit constructions in terms of lattices. In this article we generalize the flag complex description to twin cities, a generalization of twin buildings adapted to analytic completions of Kac-Moody groups as they appear for example in Kac-Moody geometry. We construct linear representations of twin cities as flag complexes of certain subspaces in Hilbert spaces.

math.DG↗
arXiv2010

Kac-Moody groups, analytic regularity conditions and cities

The relationship between minimal algebraic Kac-Moody groups and twin buildings is well known as is the relationship between formal completions in one direction and affine buildings. Nevertheless, as the completion of a Kac-Moody group in one direction destroys the opposite BN-pair, there exists no longer a twin building. For similar reasons, there are so far no buildings at all for analytic completions of affine Kac-Moody groups. In this article we construct a new type of twin buildings, called twin cities, that are associated to affine analytic Kac-Moody groups over the real or complex numbers. Twin cities consist of two sets of buildings. We describe applications of cities in infinite dimensional differential geometry by proving infinite dimensional versions of classical results from finite dimensional differential geometry: For example, we show that points in an isoparametric submanifold in a Hilbert space correspond to all chambers in a city. In a sequel we will describe the theory of twin cities for formal completions.

math.DG↗
arXiv2010

Kac-Moody geometry

The geometry of symmetric spaces, polar actions, isoparametric submanifolds and spherical buildings is governed by spherical Weyl groups and simple Lie groups. A natural generalization of semisimple Lie groups are affine Kac-Moody groups as they mirror their structure theory and have good explicitely known representations as groups of operators. In this article we describe the infinite dimensional differential geometry associated to Kac-Moody groups: Kac-Moody symmetric spaces, isoparametric submanifolds in Hilbert space, polar actions on Hilbert spaces and universal geometric twin buildings.

math.DG↗