Kac-Moody Groups Research Articles

Decompositions of Hyperbolic Kac–Moody Algebras with Respect to Imaginary Root Groups

We propose a novel way to define imaginary root subgroups associated with (timelike) imaginary roots of hyperbolic Kac–Moody algebras. Using in an essential way the theory of unitary irreducible representation of covers of the group SO(2, 1), these imaginary root subgroups act on the complex Kac–Moody algebra viewed as a Hilbert space. We illustrate our new view on Kac–Moody groups by considering the example of a rank-two hyperbolic algebra that is related to the Fibonacci numbers. We also point out some open issues and new avenues for further research, and briefly discuss the potential relevance of the present results for physics and current attempts at unification.

Product structure and regularity theorem for totally nonnegative flag varieties

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.We show that the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-totally nonnegative flag variety has a cellular decomposition into totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson varieties. Moreover, each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of U−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U^{-}$\\end{document} for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

Kato’s irreducibility criterion for Kac-Moody groups over local fields

In 2014, Braverman, Kazhdan, Patnaik and Bardy-Panse, Gaussent and Rousseau associated Iwahori-Hecke algebras to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we defined and studied their principal series representations. In 1982, Kato provided an irreducibility criterion for these representations, in the reductive case. We had obtained partially this criterion in the Kac-Moody case. In this paper, we prove this criterion in the Kac-Moody case.

SL2 representations and relative property (T)

We investigate relative property (T) of Kazhdan-Margulis for (SL2(R)⋉Rn,Rn), mainly where R is a discrete finitely generated commutative ring. The action of SL2(R) on Rn is defined through admissible lattices of the irreducible representations of sl2(C). Both positive and negative results are obtained. Applications of these results will be given, including the establishment of property (T) for certain Kac-Moody type groups over commutative rings.

Cluster algebra structures on Poisson nilpotent algebras

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.

Complexity of warped conformal field theory

Warped conformal field theories in two dimensions are exotic nonlocal, Lorentz violating field theories characterized by Virasoro–Kac–Moody symmetries and have attracted a lot of attention as candidate boundary duals to warped AdS_3 spacetimes, thereby expanding the scope of holography beyond asymptotically AdS spacetimes. Here we investigate WCFT_2 s using circuit complexity as a tool. First we compute the holographic volume complexity (CV) which displays a linear UV divergence structure, more akin to that of a local CFT_2 and has a very complicated dependence on the Virasoro central charge c and the U(1) Kac–Moody level parameter k. Next we consider circuit complexity based on Virasoro–Kac–Moody symmetry gates where the complexity functional is the geometric (group) action on coadjoint orbits of the Virasoro–Kac–Moody group. We consider a special solution to extremization equations for which complexity scales linearly with “time”. In the semiclassical limit (large c, k, while c/k remains finite and small) both the holographic volume complexity and circuit complexity scales with k.

Affine Kac-Moody groups and Lie algebras in the language of SGA3

In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.

Conversations with Flaschka: Kac–Moody groups and Verblunsky coefficients

In this paper two items are discussed which in one way or another originated from conversations with Hermann Flaschka and his students. The first is an application of the Toda lattice to the question of whether there exists a complex Lie group with an exponential map associated to an indefinite type Kac–Moody Lie algebra. The second concerns a new example of the Verblunsky correspondence.

A new axiomatics for masures II

Abstract Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau in order to study Kac–Moody groups over valued fields. We prove that the intersection of two apartments of a masure is convex. Using this, we simplify the axiomatic definition of masures given by Rousseau.

Entirety of certain cuspidal Eisenstein series on Kac–Moody groups

Let $G$ be an infinite-dimensional representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on $G$ induced from unramified cusp forms on finite-dimensional Levi subgroups of maximal parabolic subgroups. Under a natural condition on maximal parabolic subgroups, we prove that the cuspidal Eisenstein series are entire on the full complex plane.

Principal series representations of Iwahori–Hecke algebras for Kac–Moody groups over local fields

Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. We introduce principal series representations for these algebras. We study these representations and partially generalize Kato and Matsumoto irreducibility criteria.

Computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces

Computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces

Constructing a Lie group analog for the Monster Lie algebra

Let \(\mathfrak {m}\) be the Monster Lie algebra. We summarize several interrelated constructions of Lie group analogs for \(\mathfrak {m}\). Our constructions are analogs for \(\mathfrak {m}\) of Chevalley and Kac–Moody groups and their generators and relations.

FUNDAMENTAL GROUPS OF SPLIT REAL KAC-MOODY GROUPS AND GENERALIZED REAL FLAG MANIFOLDS

We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, because of the Iwasawa decomposition G = KAU+, the embedding K ,↪ G is a weak homotopy equivalence, in particular π1(G) = π1(K). It thus suffices to determine π1(K), which we achieve by investigating the fundamental groups of generalized ag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized ag variety is a CW decomposition- in particular, we cover the complete symmetrizable situation; furthermore, the results concerning only the structure of π1(K) actually also hold in the nonsymmetrizable two-spherical case.

Recognizing nullhomotopic maps into the classifying space of a Kac–Moody group

This paper extends certain characterizations of nullhomotopic maps between p-compact groups to maps with target the p-completed classifying space of a connected Kac–Moody group and source the classifying space of either a p-compact group or a connected Kac–Moody group. A well known inductive principle for p-compact groups is applied to obtain general, mapping space level results. An arithmetic fiber square computation shows that a null map from the classifying space of a connected compact Lie group to the classifying space of a connected topological Kac–Moody group can be detected by restricting to the maximal torus. Null maps between the classifying spaces of connected topological Kac–Moody groups cannot, in general, be detected by restricting to the maximal torus due to the nonvanishing of an explicit abelian group of obstructions described here. Nevertheless, partial results are obtained via the application of algebraic discrete Morse theory to higher derived limit calculations. These partial results show that null maps are detected by restricting to the maximal torus in many cases of interest.

DISTANCES ON A MASURE

A masure (also known as an affine ordered hovel) is a generalization of the Bruhat-Tits building that is associated with a split Kac-Moody group G over a nonarchimedean local field. This is a union of affine spaces called apartments. When G is a reductive group, is a building and there is a G-invariant distance inducing a norm on each apartment. In this paper, we study distances on inducing the affine topology on each apartment. We construct distances such that each element of G is a continuous automorphism of and we study their properties (completeness, local compactness, …).

On structure constants of Iwahori–Hecke algebras for Kac–Moody groups

We consider the Iwahori-Hecke algebra associated to an almost split Kac-Moody group $G$ (affine or not) over a nonarchimedean local field $K$. It has a canonical double-coset basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ indexed by a sub-semigroup $W^+$ of the affine Weyl group $W$. The multiplication is given by structure constants $a^ {\mathbf u}_{\mathbf w,\mathbf v}\in N=Z_{\geq0}$ : $T_{\mathbf w}*T_{\mathbf v}=\sum_{\mathbf u\in P_{\mathbf w,\mathbf v}} a^ {\mathbf u}_{\mathbf w,\mathbf v} T_{\mathbf u}$. A conjecture, by Bravermann, Kazhdan, Patnaik, Gaussent and the authors, tells that $a^ {\mathbf u}_{\mathbf w,\mathbf v}$ is a polynomial, with coefficients in $N$, in the parameters $q_{i}-1,q'_{i}-1$ of $G$ over $K$. We prove this conjecture when $\mathbf w$ and $\mathbf v$ are spherical or, more generally, when they are said generic: this includes all cases of $\mathbf w,\mathbf v\in W^+$ if $G$ is of affine or strictly hyperbolic type. In the split affine case (where $q_{i}=q'_{i}=q$, $\forall i$) we get a universal Iwahori-Hecke algebra with the same basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ over a polynomial ring $Z[Q]$; it specializes to our Iwahori-Hecke algebra when one sets $Q=q$.

A Birkhoff–Bruhat atlas for partial flag varieties

A partial flag variety PK of a Kac–Moody group G has a natural stratification into projected Richardson varieties. When G is a connected reductive group, a Bruhat atlas for PK was constructed in He et al. (2013): PK is locally modelled with Schubert varieties in some Kac–Moody flag variety as stratified spaces. The existence of Bruhat atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties.A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac–Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff–Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the J-Schubert varieties for a Birkhoff–Bruhat atlas. The notion of the J-Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac–Moody group). The main result of this paper is the construction of a Birkhoff–Bruhat atlas for any partial flag variety PK of a Kac–Moody group. We also construct a combinatorial atlas for the index set QK of the projected Richardson varieties in PK. As a consequence, we show that QK has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams (2007) in the case where the group G is a connected reductive group.

E 9 exceptional field theory. Part II. The complete dynamics

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E9 exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends {mathfrak{e}}_9 by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity.

Finiteness Theorems for Kac–Moody Groups over Non-archimedean Local Fields

Abstract We prove the finiteness of formal analogs of the spherical function (Spherical Finiteness), the ${\textbf c}$-function (Gindikin–Karpelevich Finiteness), and obtain a formal analog of Harish-Chandra’s limit (Approximation Theorem) relating spherical and ${\textbf c}$-function in the setting of $p$-adic Kac–Moody groups. The finiteness theorems imply that the formal analog of the Gindikin–Karpelevich integral is well defined in local Kac–Moody settings. These results extend Braverman–Garland–Kazhdan–Patnaik’s affine Gindikin–Karpelevich finiteness theorems from [ 4] and provide an algebraic analog of the combinatorial results of Gaussent–Rousseau [10] and Hébert [14].