Semisimple Group Research Articles

Unit groups of semisimple group algebras of groups up to order 180

Unit groups of semisimple group algebras of groups up to order 180

Unit groups of semisimple group algebras of groups up to order 180

In this paper, we discuss the structure of the unit groups of the semisimple group algebras of 10 non-metabelian group having orders 150, 160, 168 and 180, respectively. In particular, we consider the only one non-metabelian group of orders 150 and 180, two nonmetabelian group of order 160 and six non-metabelian group of order 168 and study the unit groups of their corresponding semisimple group algebra. With this paper, the study of the unit groups of semisimple group algebra of all the groups up to order 180 is completed.

Satake-Furstenberg compactifications and gradient map

Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and let $\tau$ be an irreducible representation of $G$ on a complex vector space $V$. Let $h$ be a Hermitian scalar product on $V$ such that $\tau(G)$ is compatible with respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by $\mu_{\mathfrak p}:\mathbb P(V) \longrightarrow \mathfrak p$ the $G$-gradient map and by $\mathcal O$ the unique closed orbit of $G$ in $\mathbb P(V)$, which is a $K$-orbit, contained in the unique closed orbit of the Zariski closure of $\tau(G)$ in $\mathrm{U}(V,h)^{\mathbb C}$. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of $G$ induced by $\tau$ are completely determined by the facial structure of the polar orbitope $\mathcal E=\mathrm{conv}(\mu_{\mathfrak p} (\mathcal O))$. Moreover, any parabolic subgroup of $G$ admits a unique closed orbit which is well-adapted to $\mathcal O$ and $\mu_{\mathfrak p}$ respectively. These results are new also in the complex reductive case. The connection between $\mathcal E$ and $\tau$ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure $\gamma$ on $\mathcal O$, we construct a map $\Psi_\gamma$ from the Satake compactification of $G/K$ associated to $\tau$ and $\mathcal E$. If $\gamma$ is a $K$-invariant measure then $\Psi_\gamma$ is an homeomorphism of the Satake compactification and $\mathcal E$. Finally, we prove that for a large class of measures the map $\Psi_\gamma$ is surjective.

A geometric approach to the Yang-Mills mass gap

I provide a new idea based on geometric analysis to obtain a positive mass gap in pure non-abelian renormalizable Yang-Mills theory. The orbit space, that is the space of connections of Yang-Mills theory modulo gauge transformations, is equipped with a Riemannian metric that naturally arises from the kinetic part of reduced classical action and admits a positive definite sectional curvature. The corresponding regularized Bakry-Émery Ricci curvature (if positive) is shown to produce a mass gap for 2+1 and 3+1 dimensional Yang-Mills theory assuming the existence of a quantized Yang-Mills theory on (ℝ1+2, η) and (ℝ1+3, η), respectively. My result on the gap calculation, described at least as a heuristic one, applies to non-abelian Yang-Mills theory with any compact semi-simple Lie group in the aforementioned dimensions. In 2+1 dimensions, the square of the Yang-Mils coupling constant gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} has the dimension of mass, and therefore the spectral gap of the Hamiltonian is essentially proportional to gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} with proportionality constant being purely numerical as expected. Due to the dimensional restriction on 3+1 dimensional Yang-Mills theory, it seems one ought to introduce a length scale to obtain an energy scale. It turns out that a certain ‘trace’ operation on the infinite-dimensional geometry naturally introduces a length scale that has to be fixed by measuring the energy of the lowest glu-ball state. However, this remains to be understood in a rigorous way.

Quantum Toda lattice: a challenge for representation theory

Quantum Toda lattice may solved by means of the Representation Theory of semisimple Lie groups, or alternatively by using the technique of the Quantum Inverse Scattering Method. A comparison of the two approaches sheds a new light on Representation Theory and leads to a number of challenging questions.

Remarks on the inverse Galois problem over function fields

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of ℓ-adic Galois representations valued in a semisimple group G using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density using a classical result of Larsen on Galois images for compatible systems.

Spin(8,C)-Higgs pairs over a compact Riemann surface

Abstract Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 , G G be a semisimple complex Lie group and ρ : G → GL ( V ) \rho :G\to {\rm{GL}}\left(V) be a complex representation of G G . Given a principal G G -bundle E E over X X , a vector bundle E ( V ) E\left(V) whose typical fiber is a copy of V V is induced. A ( G , ρ ) \left(G,\rho ) -Higgs pair is a pair ( E , φ ) \left(E,\varphi ) , where E E is a principal G G -bundle over X X and φ \varphi is a holomorphic global section of E ( V ) ⊗ L E\left(V)\otimes L , L L being a fixed line bundle over X X . In this work, Higgs pairs of this type are considered for G = Spin ( 8 , C ) G={\rm{Spin}}\left(8,{\mathbb{C}}) and the three irreducible eight-dimensional complex representations which Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) is described.

ON MODEL-THEORETIC CONNECTED GROUPS

AbstractWe introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

The uncertainty principle generalization for a connected semi-simple Lie group

In this paper, we are establishing the generalized variant of the Hardy theorem, which states that for [Formula: see text] a separable Hilbert space, [Formula: see text] a connected semi-simple Lie group with a Haar measure [Formula: see text] a unitary representation of [Formula: see text] in [Formula: see text], and [Formula: see text] a dual measure on [Formula: see text], if inequalities [Formula: see text] and [Formula: see text] hold for [Formula: see text], then the function [Formula: see text] must equal zero almost for all [Formula: see text].

Aspects of convergence of random walks on finite volume homogeneous spaces

We investigate three aspects of weak* convergence of the n-step distributions of random walks on finite volume homogeneous spaces of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesàro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesàro convergence towards Haar measure for uniquely ergodic random walks.

Dichotomy and Measures on Limit Sets of Anosov Groups

Abstract Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$, we show that a $\Gamma $-conformal measure is supported on the limit set of $\Gamma $ if and only if its dimension is $\Gamma $-critical. This implies the uniqueness of a $\Gamma $-conformal measure for each critical dimension, answering the question posed in our earlier paper with Edwards [13]. We obtain this by proving a higher rank analogue of the Hopf–Tsuji–Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups.

On the normal complement problem of finite group algebra

Let [Formula: see text] be a group, [Formula: see text] be a finite field with a positive characteristic [Formula: see text], then [Formula: see text] is the group algebra of a group [Formula: see text] over [Formula: see text] with a positive characteristic [Formula: see text]. In this paper, the normal complement problem on semisimple group algebra [Formula: see text] is examined. The normal complement problem of groups of order up to 16 in their corresponding unit groups [Formula: see text] has already been discussed. We have proved that up to isomorphism all the three non-abelian groups of order 18 and up to isomorphism all three non-abelian groups of order 20 do not have normal complement in their corresponding unit groups [Formula: see text].

Wilson lines and their Laurent positivity

For a marked surface Sigma and a semisimple algebraic group G of adjoint type, we study the Wilson line morphism g_{[c]}:{mathcal {P} }_{G,Sigma } rightarrow G associated with the homotopy class of an arc c connecting boundary intervals of Sigma , which is the comparison element of pinnings via parallel-transport. The matrix coefficients of the Wilson lines give a generating set of the function algebra mathcal {O}({mathcal {P} }_{G,Sigma }) when Sigma has no punctures. The Wilson lines have the multiplicative nature with respect to the gluing morphisms introduced by Goncharov–Shen [18], hence can be decomposed into triangular pieces with respect to a given ideal triangulation of Sigma . We show that the matrix coefficients c_{f,v}^V(g_{[c]}) give Laurent polynomials with positive integral coefficients in the Goncharov–Shen coordinate system associated with any decorated triangulation of Sigma , for suitable f and v.

Six dimensional homogeneous spaces with holomorphically trivial canonical bundle

We classify all the 6-dimensional unimodular Lie algebras g admitting a complex structure with non-zero closed (3,0)-form. This gives rise to 6-dimensional compact homogeneous spaces M=Γ﹨G, where Γ is a lattice, admitting an invariant complex structure with holomorphically trivial canonical bundle. As an application, in the balanced Hermitian case, we study the instanton condition for any metric connection ∇ε,ρ in the plane generated by the Levi-Civita connection and the Gauduchon line of Hermitian connections. In the setting of the Hull-Strominger system with connection on the tangent bundle being Hermitian-Yang-Mills, we prove that if a compact non-Kähler homogeneous space M=Γ﹨G admits an invariant solution with respect to some non-flat connection ∇ in the family ∇ε,ρ, then M is a nilmanifold with underlying Lie algebra h3, a solvmanifold with underlying algebra g7, or a quotient of the semisimple group SL(2, C). Since it is known that the system can be solved on these spaces, our result implies that they are the unique compact non-Kähler balanced homogeneous spaces admitting such invariant solutions. As another application, on the compact solvmanifold underlying the Nakamura manifold, we construct solutions, on any given balanced Bott-Chern class, to the heterotic equations of motion taking the Chern connection as (flat) instanton.

Calorons and Constituent Monopoles

We study anti-self-dual Yang–Mills instantons on {mathbb {R}}^{3}times S^{1}, also known as calorons, and their behaviour under collapse of the circle factor. In this limit, we make explicit the decomposition of calorons in terms of constituent pieces which are essentially charge 1 monopoles. We give a gluing construction of calorons in terms of the constituents and use it to compute the dimension of the moduli space. The construction works uniformly for structure group an arbitrary compact semi-simple Lie group.

Crystal limits of compact semisimple quantum groups as higher-rank graph algebras

Abstract Let O q ⁢ [ K ] \mathcal{O}_{q}[K] be the quantized coordinate ring over the field C ⁢ ( q ) \mathbb{C}(q) of rational functions corresponding to a compact semisimple Lie group 𝐾, equipped with its ∗-structure. Let A 0 ⊂ C ⁢ ( q ) {\mathbf{A}_{0}}\subset\mathbb{C}(q) denote the subring of regular functions at q = 0 q=0 . We introduce an A 0 \mathbf{A}_{0} -subalgebra O q A 0 ⁢ [ K ] ⊂ O q ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]\subset\mathcal{O}_{q}[K] which is stable with respect to the ∗-structure and which has the following properties with respect to the crystal limit q → 0 q\to 0 . The specialization of O q ⁢ [ K ] \mathcal{O}_{q}[K] at each q ∈ ( 0 , ∞ ) ∖ { 1 } q\in(0,\infty)\setminus\{1\} admits a faithful ∗-representation π q \pi_{q} on a fixed Hilbert space, a result due to Soibelman. We show that, for every element a ∈ O q A 0 ⁢ [ K ] a\in\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] , the family of operators π q ⁢ ( a ) \pi_{q}(a) admits a norm limit as q → 0 q\to 0 . These limits define a ∗-representation π 0 \pi_{0} of O q A 0 ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] . We show that the resulting ∗-algebra O ⁢ [ K 0 ] = π 0 ⁢ ( O q A 0 ⁢ [ K ] ) \mathcal{O}[K_{0}]=\pi_{0}(\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]) is a Kumjian–Pask algebra, in the sense of Aranda Pino, Clark, an Huef and Raeburn. We give an explicit description of the underlying higher-rank graph in terms of crystal basis theory. As a consequence, we obtain a continuous field of C * C^{*} -algebras ( C ⁢ ( K q ) ) q ∈ [ 0 , ∞ ] (C(K_{q}))_{q\in[0,\infty]} , where the fibres at q = 0 q=0 and ∞ are explicitly defined higher-rank graph algebras.

On the multiplicity spaces of spherical subgroups of minimal rank

AbstractLet be a complex connected semisimple Lie group and be a connected semisimple subgroup of . Consider the branching problem of decomposing the simple ‐representations as a sum of simple ‐representations . When , it is the tensor product decomposition. The multiplicity space satisfies , where the sum runs over the isomorphism classes of simple ‐representations. In the case when is spherical of minimal rank, we describe as the intersection of kernels of powers of root operators in some weight space of the dual space of . When , we recover by geometric methods a well‐known result.

Strongly dense free subgroups of semisimple algebraic groups II

It was shown in [9] that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show that the same is true for as long as the transcendence degree of the field is at least 1 in characteristic 0 and at least 2 in positive characteristic. We also consider related questions for surface groups.

Homogeneous spaces with invariant Koszul-Vinberg structures

Let M be a manifold endowed with a flat torsionless connection ∇. A symmetric bivector field h on (M,∇) is said to be a Koszul-Vinberg (K-V for short) bivector field if it satisfies: (∇α#h)(β,γ)=(∇β#h)(α,γ), where α#:=h#(α) and h#:T⁎M→TM is the contraction of h. The pair (∇,h) is called a K-V structure and the triple (M,∇,h) is called a K-V manifold. When h is non-degenerate, (M,∇,h−1) is a pseudo-Hessian manifold, otherwise, we have a singular foliation defined by Imh# called the associated affine foliation whose leaves are pseudo-Hessian manifolds. In this paper, we study invariant K-V structures on homogeneous spaces. More precisely, we give an algebraic characterization of these structures in the same spirit of Nomizu's theorem on invariant connections on homogeneous spaces, and we give many classes of examples mainly for reductive or symmetric pairs (G,H). We establish many properties of pseudo-Hessian homogeneous manifolds, in particular, we show that when G is a semi-simple Lie group then G/H does not admit any non trivial G-invariant pseudo-Hessian structure. Finally, we show that the leaves of the affine foliation associated to an invariant K-V structure are homogeneous pseudo-Hessian manifolds.

Hecke category actions via Smith–Treumann theory

Let $\textbf {G}$ be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block $\operatorname {Rep}_0(\textbf {G})$ of $\operatorname {Rep}(\textbf {G})$ by wall-crossing functors. This action was conjectured to exist by Riche and Williamson. Our method uses constructible sheaves and relies on Smith–Treumann theory.