Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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3
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810 views

Closed subgroups of GL(n)

Let's say I want to prove that a closed subgroup of GL(n,R) or GL(n,C) is a Lie group, with an atlas given by exponential of matrices (restricted to an appropriate subalgebra of gl(n)), without using ...
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39 views

On Lie theory with special functions. [on hold]

I research in Lie theory with special functions. But I saw a lot of research on the use of lie theory in hyper-geometric and hermit and other .. Is there a new kind of functions, not considered ...
2
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100 views

Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation: $$ g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F). $$ Here, ...
1
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2answers
113 views

Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
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111 views

Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation $$ A_1 +...+A_n ...
16
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1answer
2k views

Is the Duflo polynomial conjecture open?

Let $G/K$ be a symmetric space. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition, with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant ...
3
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75 views

Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations: $[U_i, U_j] = 0$ for $|i-j|>1$ ...
2
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2answers
170 views

odd length Chevalley relations (in rank two)

The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: ...
2
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0answers
81 views

Lie Symmetries of the Bessel Differential Equation

The Bessel differential equation has an arbitrary looking form, but a lot is known about it. $$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$ Is there a way to derive the Bessel ...
11
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1answer
388 views

Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?

$\newcommand{\til}{\tilde}$ Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds. Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
2
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0answers
60 views

Any bi-invariant distance on a group is inverse-invariant? [migrated]

$\newcommand{\inv}{\text{inv}}$ Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an ...
1
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1answer
76 views

Concept of Facets in the structure of reductive algebraic groups

Where can I find a precise definition of Facet ? In some online notes it is stated that Facet is a maximal subset of co-characters having the same sign for every root. But shouldn't then every facet ...
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0answers
38 views

Generalized Gaussian Decomposition

Let $G$ be a connected complex semisimple Lie group. Let $H$ be a maximal torus of $G$, let $W$ be the Weyl group of $G$, and let $N_\pm$ be a pair of opposite maximal unipotent subgroups. For each ...
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0answers
140 views

Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
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2answers
193 views

Quasi-isometric rigidity of Nil

Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...
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2answers
613 views

Solving equations in SO(3) : an open problem by Jan Mycielski

I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, ...
16
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5answers
524 views

Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?

Let $G$ be a simple algebraic group group over $\mathbb C$. Let $V$ be a self-dual representation of $G$. Let $\lambda$ be the highest weight of $V$. Write $\lambda$ as a sum of fundamental weights: ...
7
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1answer
294 views

Intuition for the Cartan connection and “rolling without slipping” in Cartan geometry

Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$. The Cartan connection is supposed to formalize what it means to "roll ...
9
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3answers
237 views

Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R). The idea is that each non-trivial ...
13
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3answers
271 views

How stable is the top cell of a Lie group?

It is well known that the fundamental class of a compact Lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable ...
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0answers
52 views

Kirillov orbit Method for Complex nilpotent groups

Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...
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78 views

Samelson Products in $SO(n)$

Given a topological group $G$ one forms the commutator $c\colon G\times G\rightarrow G$, $(x,y)\mapsto xyx^{-1}y^{-1}$. This map then factors through the smash $G\wedge G$. This map is the most ...
3
votes
1answer
174 views

Totally geodesic subgroups in Lie groups

Let $G$ be a Lie group with a left invariant metric $g$. Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every ...
6
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127 views

An analogue of Deligne--Lusztig theory for real groups?

I am considering the following analogue of Deligne--Lusztig theory: Take e.g. $G=GL_n(\mathbb{C})$, and let $F$ be the complex conjugate, then we have $G^F=GL_n(\mathbb{R})$. Consider the ``Lang ...
15
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5answers
4k views

Textbook or lecture notes in topological K-Theory

I am looking for a good introductory level textbook (or set of lecture notes) on classical topological K-Theory that would be suitable for a one-semester graduate course. Ideally, it would require ...
4
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1answer
138 views

Lie functor preserves “surjections” in synthetic differential geometry?

In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism. As pointed out ...
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13answers
17k views

why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
4
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176 views

Milnor's model of $EG$ and Kac-Moody groups

I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 ...
3
votes
1answer
82 views

Weingarten function for unitary group

Studying integration over unitary group I came across this function, the Weingarten function Wg, such that $$ \int_{\mathcal{U}(N)} \prod_{k=1}^{n} U_{i_kj_k} U^*_{m_k r_k} dU=\sum_{\tau,\sigma\in ...
8
votes
2answers
454 views

Interpret Fourier transform as limit of Fourier series

Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n ...
11
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3answers
494 views

on the center of a Lie group

I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles... First some definitions: • Let $G$ be compact, simple, and simply ...
10
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2answers
226 views

Can one describe the multiplication of two Bruhat cells?

For $G$ a simple linear algebraic group and $B$ a fixed Borel subgroup, we have the Bruhat decomposition $G = \coprod_{w \in W} B\dot{w}B$, where $W$ is the Weyl group and $\dot{w}$ is any ...
0
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1answer
81 views

Maximal split torus of universal chevalley group

Let $G$ be simply connected chevalley group over a field $K$. I am following the notations as in 'Lectures on Chevalley group' by Steinberg (Yale lectures). Let $H$ be the subgroup generated by ...
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63 views

Hessian of composite function

Given a (globally, but not necessarily locally) surjective smooth map: $V:\mathbb{R}^n \rightarrow SU(4)$ (with $n >> 4$) and the function $J_{G}: SU(4) \rightarrow \mathbb{R}$ defined by: ...
12
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4answers
629 views

Occurrence of the trivial representation in restrictions of Lie group representations

Suppose $G$ is a semisimple group, and $V_{\lambda}$ is an irreducible finite-dimensional representation of highest weight $\lambda$. Suppose $H \subset G$ is a semisimple subgroup. What is the ...
5
votes
1answer
117 views

The action of $GL_{\infty}$ on the infinite wedge space

This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya. Consider the following objects: the ...
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44 views

Are singular critical points isolated for control systems on comapct semi-simple Lie groups

Given a control system on $SU(n)$ (or any other compact, semi-simple Lie group I suspect) of the form: $\frac{d U_t}{dt} = (A + w(t)B)U_t$ where $A,B \in \mathfrak{su}(n)$ generate the algebra (to ...
4
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0answers
108 views

When can a locally compact group be approximated by discrete subgroups?

This question is about partitioning a (locally) compact group into cells by using discrete subgroups. Let $G$ be a locally compact group. (I am really most interested in the case where $G$ is a ...
9
votes
5answers
489 views

Lie groups vs Lie monoids

Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie grousp but with the axiomatic existence ...
3
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1answer
82 views

Characterization of $L^1(\text{SL}(3,\mathbb{R}))$ [closed]

Is there a characterisation of the integrable functions on SL($3,\mathbb{R}$) or an explicit expression for the Haar measure?
5
votes
2answers
124 views

Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?

Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is ...
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0answers
105 views

Poincaré inequality for connected Lie groups

Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is: symmetric, adapted (in the sense that there is no proper subgroup $H$ such ...
2
votes
1answer
193 views

Euler-Poincaré equations with constraints

It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} ...
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1answer
295 views

A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows; Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and ...
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2answers
179 views

How to “lift” a transitive group action on a manifold?

Let $M=G/H$ be a homogeneous manifold, with $G$ connected Lie group. Suppose that $\widetilde{M}$ is a covering of $M$. QUESTION: is there a general prescription to obtain a Lie group ...
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250 views

The geometry of the holomorph of a Lie group

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$) Is Hol$(G)$ always a Lie group? If the answer is yes our main questions: 1.For a left ...
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5answers
1k views

Which norms have rich isometry groups?

Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of all bijections ...
2
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165 views

Manifolds as simultaneous coset spaces

Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of ...
12
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2answers
2k views

Left and right eigenvalues

A quaternionic matrix $A$ gives rise to a function $\mathbb{H}^n \to \mathbb{H}^n$ given by $x \mapsto A \cdot x$. This is real linear, but not complex- or quaternionic-linear (in general) if we ...
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G-invariant functions on manifold for G compact

In a paper I saw the following statement: Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...