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Questions tagged [lie-groups]

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 (group-theory) tag.

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Are there non-embedded tori in Lie groups?

Lie subgroups are certainly not always embedded (there is the example of the $\mathbb{R} \to S^1 \times S^1$ given by a line of irrational slope). Can you have a torus that is a subgroup of a Lie ...
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Show that $\mathfrak{b}_3 (\mathbb{C}) / \mathfrak{n}_3 (\mathbb{C}) \cong \mathfrak{t}_3 (\mathbb{C})$

Show that $\mathfrak{b}_3 (\mathbb{C}) / \mathfrak{n}_3 (\mathbb{C}) \cong \mathfrak{t}_3 (\mathbb{C})$, in which $\mathfrak{b}_3 (\mathbb{C}),\mathfrak{n}_3 (\mathbb{C}),\mathfrak{t}_3 (\mathbb{C})$ ...
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Getting representations of the Lie group out of representations of its Lie algebra

This is something that is usually done in QFT and that bothers me a lot because it seems to be done without much caution. In QFT when classifying fields one looks for the irreducible representations ...
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Covering map of the universal cover $\widetilde{G} \rightarrow G$ for $G$ a Lie group is a homomorphism?

In a paper I'm reading, we have a compact Lie group $G$ and he says "We can identify $\pi_1(G)$ with the kernel of $\widetilde{G} \rightarrow G$. I can't seem to find anything that says that the ...
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Lie algebras of infinite dimensional Lie groups

I have to work with Lie algebras of some infinite dimensional 'Lie groups' (e.g. $\Omega SL_2(\mathbb{C})$) but i'm not sure on how to approach infinite dimensional groups, for loop group it is not so ...
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What is the notation $\delta$ here?

I'm reading the book "Introduction to Lie Algebra and Representation Theory - J. E. Humphrey", I have a question on an example on the page number $2$. That is Example: For reference, we write down ...
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Are algebraic groups defined over $\mathbb{R}$ Lie groups?

In the notes I am reading, which is about algebraic groups, in the section about over $\mathbb{R}$ all the sudden they started using the word Lie groups. I understand Lie groups and algebraic groups ...
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$SL_2 (\Bbb R) × SL_2 (\Bbb R)/ ± (I_2 , I_2 ) → (SO_{2,2})^\circ$

I went through this problem in Lie groups: i) Prove that $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ is a linear Lie group. I identified $SL_2 (\Bbb R) × SL_2 (\Bbb R)$ with $\{\begin{pmatrix} A & 0 \\ 0 &...
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$\operatorname{coker}(\phi)$ is discrete for a morphism of Lie groups

suppose $\phi: G \to H$ a morphism of Lie groups such that $d\phi$ is surjective. Prove that $\operatorname{coker}\phi$ is discrete. My attempts: Prove that $\phi(G)$ is open which will lead to $H/\...
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Rigorous proof of one-dimensional Lie subgroup existence.

So, that's an exert from Hochschild's "The structure of Lie Groups" p.79 Existence and uniqueness of one-parametric Lie-Group embedding such that the given tangent vector of $G$ is the image of the ...
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Relation between Symmetric algebra and Universal enveloping algebra as Lie algebras.

Let $L$ be a Lie algebra over $\mathbb{C}$. Assume $L$ satisfies PBW theorem. We can associate two Lie algebras with $L$: 1) $U(L):$ the universal enveloping algebra. Here the Lie bracket is defined ...
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Faithful representation of $C_c(X \times X)$.

Let $X$ be a smooth manifold. $X \times X$ is product manifold. $\mu$ is a Borel measure on $X$. There are two aims (i) Associate a $C^*$ norm to $C_c(X\times X)$, making it a $C^*$ algebra. (ii)...
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When is exponential map from Lie algebra to Lie group a covering map?

Suppose $G$ is a Lie group and $\mathfrak{g}$ its Lie algebra. It is not so difficult to see that if $G$ is abelian and connected then $\exp:\mathfrak{g}\rightarrow G$ is a universal covering map. ...
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Singular points of a matrix when the entries are restriced to a Lie Group

Let $\mathsf{SO}(3)$ be the set of $3 \times 3$ rotation matrices. Let $R\in\mathsf{SO}(3)$ and $r_{ij}$ represent the entry of $R$ sitting at the $i^{th}$ row and $j^{th}$ column, i.e., $$ R \in\ \...
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relation between trace and hat operator (skew-symmetric matrices)

To avoid confusion, let me first introduce the notation (although pretty standard) which is required for the question that I want to ask. Let $\mathsf{GL}(3,\mathbb{R})$ be the set of $3\times 3$ real ...
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Semisimple linear algebraic group

Let $G$ be a linear reductive algebraic over an algebraically closed field of characteristic $0$. I know that there is a surjective map with finite Kernel $$G' \times T \to G$$ where $G'$ is ...
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Generating set of lie algebra su(3)

I am looking for (an example of) a minimal set of Gell-Mann matrices such that their closure under the Lie bracket is all of $\mathfrak{su}(3)$. By minimal I mean the set should be as small as ...
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commutator ideal for direct sum

Is commutator ideal compatible with direct sum? Let's take $\mathfrak{sl}_2(\Bbb K)\oplus\mathfrak{sl}_2(\Bbb K)$ which is semi-simple because $\mathfrak{sl}_2(\Bbb K)$ is simple Lie algebra. So we ...
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Reduced group $C^*$ algebra inequality

This is part of the definition of constructing the reduced $C$-norm. Let $G$ be a locally compact hausdorff group, $\nu$ a Haar measure that is both left and right invariant, $\xi\in B(L^2(G))$, ...
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$K$-finite vectors are smooth: existence of $K$-finite approximations of the identity

Let $K$ be a compact real Lie group. Is it true that for every neighborhood $U$ of the identity in $K$ there exists a nonzero smooth function $f : K \to \mathbb C$ supported in $U$ such that $f$ is $K$...
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Is the Unitary Group of a Hilbert Space a Lie group?

Let $H$ be an infinite-dimensional complex Hilbert space. Then the set of unitary operators on $H$ forms a group, known as the unitary group or Hilbert group. My question is, is this group a Lie ...
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Adjoint representation of $\mathfrak{b}_2$ is undecomposable

Let $\mathfrak{g}=\mathfrak{b}_2(\Bbb C)$. Prove that adjoint representation $ad_\mathfrak{g}$ of $\mathfrak{g}$ is undecomposable into a direct sum of irreducible representations. My attempt: I ...
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adjoint representation is irreducible iff $\mathfrak{g}$ is simple

I am trying to prove that for a Lie algebra $\mathfrak{g}$: $ad_{\mathfrak{g}}$ the adjoint representation of $\mathfrak{g}$ is irreducible iff $\mathfrak{g}$ is simple. I tried to use the fact that ...
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Is $SL(n,\mathbb{R})/SL(n, \mathbb{Z})$ a Hausdorff space?

The special linear group $SL(n, \mathbb{R})$ of degree $n$ over $\mathbb{R}$ is the set of $n \times n$ matrices with determinant $1$, with the group operations of ordinary matrix multiplication and ...
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Simple Lie algebra representations and tensor powers of fundamental representations [duplicate]

Let $\frak{g}$ be a simple Lie algebra over $\mathbb{C}$. We will call a representation of $\frak{g}$ tautological if it is a fundamental representation of smallest dimension. For $V$ a tautological ...
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If $X,Y$ generates $\mathfrak{g}$ then $e^{tX}$ and $e^{tY}$ generates de Lie Group G.

I'm trying to solve the following problem of the book "Grupos de Lie - Luiz A. B. San Martin": Question: Let $G$ be a connected Lie Group with Lie algebra $\mathfrak{g}$. Suppose that $X,Y \in \...
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Characters of the fundamental representations of $SU(3)$

Let us denote $3$ and $\bar{3}$ the fundamental representations of $SU(3)$. According to my lecture notes, the characters read as follows: $\chi_{[3]} = e^{\omega_1} + e^{\omega_1 - \alpha_1} + e^{\...
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Lie group-algebra representations

I want to prove the following: Given two representations of a connected matrix Lie group are equivalent if and only if the associated Lie algebra representations are equivalent. Definition: Let $G$ ...
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If two sub-algebras $\mathfrak{h}, \mathfrak{h}'$ generate $\mathfrak{u}(n,\mathbb{C})$, their Lie groups generate $U(n,\mathbb{C})$.

I'm working my way through this paper: https://arxiv.org/pdf/quant-ph/0108062.pdf On page 4, in the fourth step of the outline of their argument, they make the following claim without proof: If $H, ...
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The representation $\Phi_{n} $ of $SU_{2}$ is irreducible.

The proof is given below: But I do not understand why " we first determine which of the spaces of $V_{n}$ are invariant under T " as he said in the second sentence. could anyone explain this for ...
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A discrepancy in a paragraph in the construction of a series of irreducible complex representations of $SU_{2}$

The construction is given below: But I do not understand the last sentence and the paragraph before it, Could anyone give me a concrete example to explain them please? Thank you!
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$U_{p,q}$ is bounded

I am trying to prove $U_{p,q}$ is bounded using the induced norm $|| . ||_2$ from $M_n(\Bbb R)$ (or $M_n(\Bbb C)$ I am not sure). A norm is an application $M_n(\Bbb C) \to \Bbb R^+$, but in the case ...
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Left translation on Lie group of a discrete subgroup is properly discontinuous

This question has been asked before here and there but has not received answers which make clear my difficulties understanding this argument. I am quite rusty in both group theory and topology, and I ...
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How to show a real matrix $A$ belongs to Indefinite Orthogonal Group $O(n;k)$?

I want to show that an $(n + k) \times (n + k)$ real matrix $A$ belongs to $O(n;k)$ iff $gA^Tg = A^{-1}$. I know that for all $\vec{x}, \vec{y} \in \mathbb{R}^{n+k}$, and for the matrix $$ g = \...
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Linearization of the Pythagorean theorem with matrices

Suppose the differential relation $ds=\alpha dx+\beta dy$. Squaring each side of the relation, we obtain: $$ (ds)^2=\alpha^2(dx)^2+\beta^2(dy)^2+\alpha\beta dxdy+\beta\alpha dydx $$ The structure of ...
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How is the Lie bracket defined for two elements in the dual (cotangent) space?

If $\xi\in se(3)$ and $\xi^{*}\in se^{*}(3)$. How can we define $[\xi^{*}_{1},\xi^{*}_{2}]$?
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Affine Group Scheme Definition

From what I understand, a affine group scheme $G$ should be an affine scheme on which there exists a group structure in the sense that $$ \phi: G \times_k G \to G $$ is also a morphism of groups. ...
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Reducing the Dimensionality of the Sphere in terms of the Lie Algebra

The $n$-sphere can be written as an $(n-1)$-sphere fibered over an interval $$ ds^2_{\Omega_n} = d\theta^2 + \sin^2 \theta\;d\Omega_{n-1}^2. $$ In these coordinates, when we impose that we keep $\...
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Are these enough conditions for the subgroup to be a full latice?

I was wondering how to prove the following, or if you like, whether it is true, although I am almost certain it is. Let $V$ be a finite dimensional vector space over $\mathbb{C}$, say of dimension $n$....
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1answer
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$Ker(exp_G )$ is discrete

$G ⊂ GL_n (\Bbb R)$ is an abelian connected Lie group, $\mathfrak{g}$ its Lie algebra and $exp_G : \mathfrak{g} → G$ the exponential map. Prove that $Ker(exp_G )$ is discrete. My attempt: $Lie(Ker(...
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Deducing the additive Jordan decomposition

In $M_n(\Bbb C)$, I could prove that the additive Jordan decomposition of $X=D+N$ with $D$ diagonalizable and $N$ nilpotent gives a multiplicative Jordan decomposition $e^X=e^De^N$. Is that true the ...
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dφ is bijective but φ is not a lie group isomorphism

suppose G and H connected Lie groups. Is there $\phi: G \to H$ a morphism of Lie groups such that $d\phi$ is bijective but $\phi$ is not an isomorphism of Lie groups? I know that $d\phi$ surjective ...
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Expressing every element of a lie group as a product of exponentials?

Given a connected lie group $G$, since a neighbourhood of the origin generates all of $G$, we have that every $g$ in $G$ can be expressed as a finite product of elements of the form $e^X$, for $X$ in ...
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induced morphism between lie groups is surjective

Let $φ : G → H$ be a continous morphism of linear Lie groups with H connected. Prove that $φ$ is surjective iff $dφ$ surjective Using the expression : $exp\circ dφ = φ \circ exp$ we can see that $...
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Compact sympletic group

I was reading Brian C hall book on Lie algebra In that I come across following I had following with me But form this I can not conclude highlighted text . Please Help me . Any Help will be ...
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1answer
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Dimension of image of one parameter subgroup. [closed]

If $G$ is a Lie group then $\eta : \mathbb{R}\to G$ is called one parameter subgroup if it is a continuous group homomorphism. I need to show that the images of one-parameter subgroups in a Lie group ...
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Show that every irreducible representation of $SO_{3}$ is isomorphic to one of the representations $\Psi_{n}$.

The question is given below: And this is the mentioned exercise: And this is 7.4: Could anyone give me a hint about the solution of the question, I am stucked in it ?
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The automorphism group of a Lie algebra

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra. It is well-known that the the automorphism group of $\mathfrak{g}$, $\operatorname{Aut}(\mathfrak{g})$, is an algebraic group. How ...
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If $G$ is a Lie group and $H$ is a closed Lie subgroup, then $G\to G/H$ is a principal- $H$ bundle.

Let $G$ be a Lie group and $H$ be a closed Lie subgroup of $G$. Let $G/H$ has the quotient topology. Then $ p: G\to G/H$ is a principal-$H$ bundle. I was reading the above theorem from the book ...
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1answer
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Use the theory of characters to derive the following relation for the representations of $SU_{2}.$

The question is given below: And the hint at the back of the book says: Establish the corresponding equality for characters. And this was a question I was helped on it, which establish the relation ...