A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group. Consider using with the ...

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How To Formalize the Fact that $(g, h)\mapsto dL_g|_h$ is smooth where $g, h\in G$ a Lie Group

Let $G$ be a Lie group. I am wondering if there is a way to say that the map $(g, h)\mapsto dL_g|_h$ defined on $G\times G$ is a smooth map (Here $L_g$ is the left translation map from $G$ to $G$ ...
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47 views

Questions about fundamental representations of $SL_3/U$.

Consider the group $SL_3$. Let $U$ be the subgroup of $SL_3$ consisting of all upper triangular unipotent matrices. Then the algebra $\mathbb{C}[SL_3/U]$ is generated by $a_{11}, a_{21}, a_{31}, ...
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42 views

A Lie Group Homomorphism $f:G\to H$ Induces a Functor from Principal $G$-Bundles to $H$-Bundles

I am trying to understand Qiaochu Yuan's answer to this question. The first line of the answer reads: A Lie group homomorphism $f:G\to H$ induces a functor from the category of principal ...
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Defining relations of SU(2) and SU(4)

Defining relations are relations among the generators of a group, $\{g_a\}$, such that $\prod g_a^{n_a}=e$, where $e$ is the identity element. The multiplication rule of a group can always be ...
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40 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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27 views

Real version of Harish-Chandra-Itzykson-Zuber integral

I'm interested in an integral of the form $$ \int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU $$ where the integration is with respect to the Haar measure on the orthogonal group, ...
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65 views

Is an injective morphism from a Lie group to itself surjective?

I have a question about Lie groups. Let $G$ be a finite dimensional (real or complex) Lie group and $f:G \rightarrow G$ an homomorphism of Lie groups. Edit : in the view of some counter-examples let's ...
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24 views

Closure of a Manifold is a Manifold with Corners?

Is there a general theorem that shows that if you have a manifold $S$ then its closure $\overline{S}$ is a manifold with corners? I am dealing with a specific set $S$ (I would rather not say which ...
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31 views

Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
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22 views

The definition of Fell topology

Let $G$ be a Lie group, $\pi$ is a representation, then with some conditions, we have the following branching law $\pi|_N=\int^{\oplus}m_\pi(\mu)\mu\mathrm{d}\mu$ where $m$ is the multiplicity ...
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36 views

Adding tori to semi-simple groups

Let $G$ be a complex, connected, semi-simple Lie group (throw in simply connected if you like) with Lie algebra $\mathfrak g$. Let $T \subseteq B$ be a maximal torus and choice of Borel, respectively. ...
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1answer
35 views

The fundamental vector fields of a principal bundle are vertical.

Let $p:P\to M$ be a principal $G$-bundle. To each $A$ in the Lie algebra of $G$ corresponds a fundamental vector field $A^*$ on $M$ defined by $$A^*_u=\frac{d}{dt}|_{t=0} u(exp(tA))$$ How can we see ...
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30 views

Action on an irreducible lattice of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ [closed]

Let $G_1 = G_2 = \mathrm{SL}_2(\mathbb{R})$. Let $\mathrm{SL}_2(\mathbb{Z}[\sqrt{2}]) = \Gamma < G_1 \times G_2$ be the usual example of an irreducible lattice. Is the action of $G_1$ on $(G_1 ...
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1answer
47 views

Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
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3answers
103 views

Proving smoothness of left-invariant metric on a Lie Group

Assume $G$ is a Lie group. The standard construction of a left invariant metric on $G$ goes as follows: Take an arbitrary inner product $\langle,\rangle_e$ on $T_eG$ and define $\langle u , ...
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27 views

Lie algebra operations from lie group

According to wikipedia, if $G$ is a closed subgroup of $GL(n, \mathbb{R})$ then the Lie algebra of $G$ can be thought of informally as the matrices $m$ of $M(n, \mathbb{R})$ such that $1 + εm$ is in ...
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35 views

Is SU(2) a subgroup of the exceptional lie group $G_2$?

I am not an expert in Lie groups and I have spent ages looking at textbooks; I assume that because I haven't found this statement explicitly it must either be untrue or obvious ;) The only thing I ...
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1answer
26 views

Lie Derivative of Connection 1 form

On Page 106 of Kobayashi & Nomizu's 'Foundations of Differential Geometry', the authors write \begin{align*} (L_X \omega)(Y)&=X(\omega(Y))-\omega([X,Y]). \end{align*} Here, $\omega$ is the ...
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87 views

Unique metric for the Hyperbolic Half Plane Model?

I was reading today that there is a unique metric (up to multiplicative constant) that preserves distances wrt to linear fractional transformations: $$z \mapsto \frac{az + b}{cz + d}$$ of the upper ...
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37 views

Write every element of a nilpotent Lie subgroup as product of exponentials of simple generators

I have a question about Lie groups. Let $G$ be a simply connected semi-simple complex Lie group and $\mathfrak{g}$ its Lie algebra. We consider a Cartan-Weyl basis of $\mathfrak{g}$, giving the usual ...
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1answer
78 views

Groups that are not Lie Groups

What are some examples of groups that can not be given a smooth structure such that they become a Lie Group? Edit: To be a bit more specific, I was hoping that somebody could give an example of a ...
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42 views

Ideals of Lie-algebras

I am wondering whether the following claim is true: Let $G$ be a Lie group, $\mathfrak{g}$ its Lie algebra and $V$ some vector subspace of $\mathfrak{g}$. Claim: $V$ is an ideal of $\mathfrak{g}$ ...
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Are the elements of the adjoint represetnation normal operators

Given a Lie group $G$ with Lie algebra $\mathfrak{g}$ on has the adjoint action of each $g\in G$ given by $Ad_g(\mathfrak{g})$. Is $Ad_g: \mathfrak{g} \rightarrow \mathfrak{g}$ a normal operator ...
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32 views

A question about tensor products of representations

Let $V$, $W$ be real finite diemnsional vector spaces. What is the relationship between the subgroup of $GL(V\otimes W)$ whose elements can be written as tensor products (I think we can write this ...
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116 views

Lie Groups/Lie Algebra - Applications?

I studied Lie Groups and Lie Algebras as part of my Masters back in the 1970s. Although very elegant and beautiful, it seemed to its own little world, I never saw the connection with other branches of ...
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1answer
54 views

Given adjoint action find original matrix.

Given the Adjoint action of a matrix; $\text{Ad}(g) X_1 = g \, X_1 \, g^{-1} = X_2 $. Where g is in a (matrix) Lie group, $X_1,\; X_2$ are from the Lie algebra, can a $g$ be written in terms of the ...
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57 views

Characterization of Semidirect Products: Too strong assumptions?

In John Lee's book Introduction to Smooth Manifolds, there is the following Theorem. Theorem 7.35 (Characterization of Semidirect Products). Suppose $G$ is a Lie group, and $N,H\subseteq G$ are ...
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32 views

Nonlinear Lie group from Fulton & Harris

On page 138 of my copy of the celebrated Representation Theory by Fulton & Harris, a proof is outlined to show that the real group of $3\times 3$ upper-triangular unipotent matrices modulo a ...
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31 views

use of existence of bi-invariant differential form on a Lie group?

In do-carmo's Book "Riemannian Geometry" there is an exercise on proving existence of a bi-invariant metric on any compact connected Lie group. (pg 46, question 7). In the first stage, you are ...
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30 views

Uniqueness of bi-invariant metrics on Lie groups?

As noted here , a Lie group $G$ admits a bi-invariant metric if and only if $G$ is the cartesian product of a compact (Lie) group and a vector space $\mathbb{R}^n$. The question: For which Lie ...
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Is there a non-abelian Lie group which is homeomorphic to an $n$-dimensional torus $\mathbb{T}^n$?

I've learned that a compact connected abelian Lie group must be a torus. Of course, conversely, a torus as a group is abelian. I wonder if 'homeomorphic to a torus' is enough to imply abelian. ...
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1answer
37 views

A compact Lie group modulo by its maximal torus has nonzero Euler characteristic

In Andrew Baker's Matrix Groups, (in the proof of Theorem 20.11), there is an unproven statement that if $G$ is a compact Lie group and $T$ is a maximal torus, then $\chi (G/T)\ne 0$. I have an ...
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1answer
26 views

What is the automorphism group of the compact symplectic group?

I would like to know what the group of outer automorphisms of $Sp(2)$ is. I think this should be isomorphic to $\mathbb{Z}_2$, but I am not completely sure.
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1answer
41 views

Implications of $RR^T =\mathbf1$

Let $R:I→SO(3)$, smooth. We know that, for any value of $t∈I$, $R(t)R(t)^T=\mathbf1$, where $\mathbf1$ is the identity matrix. Then, differentiating both sides one finds that ...
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1answer
54 views

Quotient manifold theorem provides a fibration?

It's known that if $G$ is a Lie group and $H\subseteq G$ is a closed subgroup, then the quotient map $p\colon G\to G/H$ is in fact a principal $H$-bundle, which follows from the existence of local ...
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2answers
76 views

Lie group step in proof

Let $X_e,Y_e \in T_eG$ be vectors and $G = GL(n).$ Then the right translation is given by $Y_g = Y_e g$ and $X_g = X_e g.$ Now, I have a proof showing that $[X_e,Y_e] \in T_eG$ is the element ...
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1answer
53 views

Is GL($2$,$\mathbb{Z}$) is lie group?

This is a very dumb question, but is $\mathrm{GL}(2,\mathbb{Z})$ is lie group? I don't think it is, since its underlying set don't form a manifold, but I am just not sure.
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23 views

Lie algebra of projective linear group and one-parameter transformations

The vector fields $\partial_x$, $x\partial_x$ and $x^2\partial_x$ in $\mathbb{R}$ have correspondant flows $x\mapsto x+t$, $x\mapsto e^t x$ and $x\mapsto (1-tx)^{-1}$ which are translation, dilations ...
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1answer
92 views

Is there a compact manifold having Euler characteristic 0 which cannot be given a Lie group structure?

I realized that a (compact) Lie group must have Euler characteristic 0 due to Poincare-Hopf index theorem. Now I'm thinking of its converse. Is there a compact manifold having Euler characteristic 0 ...
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1answer
29 views

Computing plethysms of the adjoint representation using the Littlewood Richardson rule

Let $N$ be an integer (let's imagine very large), and let $G$ be the group $\mathrm{GL}_N(\mathbb{C})$. I would like to compute various plethysms of irreducible representations which are not ...
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1answer
45 views

Lie Algebra to Lie Group Mapping.

When I map a lie algebra vector in se3 to SE3 using exponentiation and map it back to se3 using log, why do I get significantly different results? I followed this and coded an implementation in ...
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34 views

Space of $G$-invariant Riemannian metrics contractible?

A well-known result in (psuedo)Riemannian geometry is that the moduli space of (pseudo)Riemannian metrics on a smooth manifold is contractible. In the case when you have a smooth action of a group $G$ ...
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20 views

Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and ...
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1answer
57 views

What is the determinant of Ad(g)?

In more generality, if a matrix acts on a group of matrices by conjugation, what is the determinant of this action (if such a notion exists)? Is it simply the determinant of the matrix being used to ...
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1answer
27 views

Compactness of a group with a bounded left-invariant metric

Let $G$ be a group equipped with a left-invariant metric $d$: that is, $(G,d)$ is a metric space and $d(xy,xz) = d(y,z)$ for all $x,y,z \in G$. Suppose further that $(G,d)$ is connected, locally ...
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1answer
25 views

Existence of the universal covering space of a connected Lie group

I am working on a project about how the universal cover of a connected Lie group is a Lie group, but I cannot find a theorem that assures that this universal cover actually exists. I've found ...
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1answer
29 views

dimension of lie algebra

I am studying lie algebra myself and question is about finding dimension of lie algebra . While i read Wikipedia link about lie algebra and lie group i saw statement Lie algebra $\mathfrak{g}$ is ...
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25 views

Connection of a $G$-principal bundle as a section of a vector bundle?

In what follows all manifolds, Lie groups and mappings are meant to be $C^\infty$. Let $\pi:M\longrightarrow B$ be a left $G$-principal bundle. A connection on this bundle is a map $H$ which assigns ...
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1answer
27 views

torus in $SU(2)$ yields a torus in $SO(3)$

in John Stillwell's book "Naive Lie Theory" there is an exercise to explain why a torus in $SU(2)$, (sub group that is isomorphic to $S^1 \times S^1$) yields a torus in $SO(3)$ (in order to prove that ...
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1answer
90 views

Non trivial homomorphism from $SU(2)$ to the diffeomorphism group of the circle

Is there a non trivial homomorphism $f: SU(2) \to \operatorname{Diff}(S^1)$? (From the comments) By a previous question, we know that there is no nontrivial homomorphism $SU(2) \to O(2)$. Since ...