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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Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
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References for metrics questions in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
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17 views

If the continuous action of a non-compact Banach-Lie group on a Banach space preserves the zero element, then it is non-proper.

I am studying the differential geometry of Banach-Lie groups, specifically, the differential geometry of the orbits of an action of a Banach-Lie group on a Banach space, and I ended up "proving" the ...
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9 views

Showing that the commutator subgroup of a Lie group is a Lie subgroup

I'm learning about Lie groups and Lie algebras independently, and I'm trying to show that the commutator subgroup, $H=[G,G]$, of a Lie group, $G$, is a Lie subgroup. My first instinct was to take a ...
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151 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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24 views

References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
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152 views

Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
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Bruhat decomposition and the order of the group.

Suppose we have a group $G(q)$ over a finite field $\mathbb{F}_q$. How can the Bruhat decomposition be used in order to calculate the order of $G(q)$? Are there any examples for some particular ...
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26 views

Finite dimensional algebraic representation of $SL_2(\mathbb{C})$

I heard that for each $n\in \mathbb{N}$, there is the unique algebraic irreducible representation of $SL_2(\mathbb{C})$ with dimension $n$ over $\mathbb{C}$. Would you let me know what is such ...
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Functoriality of the adjoint representation

Just a simply question. I came across the following statement which is used for deriving Weyl's integral formula: ''$\text{Ad}_G(h)|_{\mathfrak{h}} = \text{Ad}_H(h)$ due to functoriality in the Lie ...
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Calculating the Lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$, realized as $$SO(2,1)=\{X\in \operatorname{Mat}_3(\mathbb{R}) \,|\, X^t\eta X=\eta, \det(X)=1\},$$ where $$\eta = \left ( ...
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14 views

Show that we have a smooth path in $T_1(G)$, the tangent space of a matrix group

Consider the path $D_s(T)=A(s)B(t)A(s)^{-1}B(t)^{-1}$ in $G$ for some fixed value of s. Then the Lie bracket $[X,Y]$ can be related to the commutator of $A(s)B(t)A(s)^{-1}B(t)^{-1}$ of smooth paths ...
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31 views

Relationship between invariant and harmonic forms

Let $G$ be a compact real Lie group which is also a complex manifold (note that I do not want $G$ to be a complex Lie group). Let $\Omega^{p , q}(G)$ denote the space of smooth $(p , q)$-forms on $G$, ...
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1answer
15 views

Isotropy algebra for $U(n)$?

Let $G = U(n)$ be the Liegroup of $n \times n$ unitary matrices and $\mathfrak{g}$ the corresponding Lie algebra. Now $G$ can act on $\mathfrak{g}$ by the Adjoint-action. Since $G$ is a subgroup of ...
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208 views

Prerequisites to “Applications of Lie Groups to Differential Equations”

I'm currently a 4th year student at a university. I've become close with a professor and we talked about the topic of lie groups in differential equations. He then offered to do a reading course with ...
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45 views

Infinite Lie algebra

It is well known fact that the finite dimensional Lie algebra will always be closed under commutation relation. But I have doubt about infinite Lie algebra. In some of the case it is closed under ...
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25 views

Every orthogonal matrix represents a rotation around an axis

Is it true that every element of the group $O(n)$ represents a rotation around some axis? I'd like this to be true in order to decompose any matrix $R \in O(n)$ as a block matrix in $O(n-1)$ and a 1 ...
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22 views

SU(N) tensor product decomposition

Let's consider the group SU(N). The adjoint representation is $\textbf{Adj}= $ $\textbf{N}^2\textbf{-1}$. The following decomposition holds generally ( have a look at this ref ) $$ ...
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42 views

The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
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41 views

Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
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34 views

A semisimple Lie group has no character; Am I right?

Let $G$ be a compact connected Lie group with semisimple Lie algebra ${\frak g}$. With the following reasoning, I show that there is no non-trivial Lie group homomorphism $$\chi:G\to S^1.$$ Is that ...
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36 views

Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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19 views

Exponentiating an ``affine subalgebra''

Consider the Lie algebra $u(N)$. If I exponentiate an $x \in u(N)$, I will obtain an element of the $U(N)$ group. My understanding is that $exp(u(N))= \{exp(x)| x \in u(N)\}$ is, in fact, the group ...
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1answer
32 views

Left-invariant vector fields on the circle $S^1$

I'm trying to find the left-invariant vector fields on the circle $S^1$. If I understand correctly, $S^1$ is given the group structure of the multiplicative group of complex numbers on the unit ...
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161 views

Is geodesic distance equivalent to “norm distance” in $SL_n(\mathbb{R})$?

Take any norm, $\|\cdot\|$on $\mathbb{R}^n,$ and consider the resulting norm on $SL_n(\mathbb{R})$: $$\|A\|:= sup\{\|Av\|: \|v\|=1\}.$$ Now take any left-invariant Riemannian metric, $g$, on ...
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1answer
22 views

Coset spaces of a lie group and fiber bundles

I am reading Steenrod. He writes: Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point ...
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21 views

Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle ...
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1answer
18 views

What's its use of the nonsingular 2-step nilpotent Lie algebras

What's its use of the nonsingular 2-step nilpotent Lie algebras which form an a class of 2-step nilpotent Lie algebras ? Recall: A $2$-step nilpotent Lie algebra $N$ is non-singular if $ad X : N \to ...
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715 views

Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra. My motivation is that I (eventually) want to understand the theory underpinning papers ...
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1answer
33 views

Showing that a very well-known representation is really a representation

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to ...
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how to show that the representation of $SL(2, \mathbb{C})$ is holomorphic

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to ...
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19 views

Problems with Matrix of irreducible representation of SO(3)

I'm working out the irreducible representation of $SO(3)$. Let's call $R_{\theta}$ as a general rotation and $Y_{m}^{l}\left(\eta,\varphi\right)$ the spherical harmonics. I now like to have the matrix ...
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1answer
25 views

Classification Problems in Lie algebra

My field of interest is Lie group analysis of PDEs, currently I am struggling with techniques for classification of Lie algebra into mutually conjugate classes of 1- and 2-dimensional sub-algebras ...
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24 views

Fixed-point subspace of $O(2)^-$, a subgroup of $O(3)$

$O(2)^-$ is generated by the $SO(2)$ of rotations about the $z$-axis and a reflection through a vertical plane. The space $V_l$ is generated by spherial harmonics, i.e., Cartan decomposition ...
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62 views

Calculating the differential of the inverse of matrix exp?

Let $A(t)$ and $B(t)$ be two matrix-valued smooth function satisfying the equation, $B(t) = e^{A(t)}$. I need to express $\frac{dA(t)}{dt}$ in terms of $B(t)$. I know that there is a formula of ...
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14 views

Centralizer of Maximal Toral Subalgebra of $L$

I am trying to understand the proof of the following result from Humphreys Lie Algebra book: Let $L$ be a semisimple complex Lie Algebra.Let $H$ be Maximal toral subalgebra of $H$,Let $C_L(H)$ ...
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1answer
18 views

Can we discribe a Lie group action from some local property?

Let G be a Lie group,and it acts on a smooth manifold M.Then can we get that the action is transitive from some local property of the Lie group action.More precisely,Can we get the action is ...
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1answer
38 views

Quotient by a discrete subgroup of a Lie group

I was reading Fulton Harris' Representation theory, A first course, where I came across the following: Let $H$ be a Lie group and $T$ be a discrete subgroup of its center $Z(H)$. Then there exists ...
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1answer
12 views

Lie group structures inside of Clifford algebras

I am reading a text by Jean Gallier on Clifford algebras, Pin and Spin groups. I have a problem with one little innocent-looking paragraph establishing Pin and Spin as Lie groups on the page 37. I ...
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1answer
23 views

$S^1$ acting on $SO(n+1)/SO(n-1)$ by translations

I'm ready right now in a paper, that $S^1$ acts on $SO(n+1)/SO(n-1)$ by right translations. I thought that a Liegroup $G$ acting by right translations, means that we have a right action $\varphi ...
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1answer
23 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
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Relationship between Casimir and index of a representation of a Lie algebra.

In several QFT textbooks (namely, those of Peskin and Shroeder and of Schwartz) there is presented an identity for representations of Lie algebras, $$ d(R) C_2(R) = T(R) d(G),$$ where $d(R)$ is the ...
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1answer
43 views

Commutation relation between su(N) and clifford algebra generators.

Why does the $ \gamma_5 $ matrix commute with the generators of the $su(N)$ algebra? In the case of the chiral symmetry from physics, [$Q_a$, $Q_b^5$] = $i \epsilon_{abc}Q^5_c$ where the $Q_a, Q^5_a$ ...
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34 views

Why Bi invariant metric on noncompact lie group doesn't exist??

In the book "Lectures on Differential Geometry" by Sternberg page 233 "Given a representation,p, of a Lie group G (in particular the adjoint representation) on a vector space F, if p(x) is compact ...
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14 views

Is this covering group $G'$ of $G$ unique?

Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). ...
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16 views

In what condition, the universal covering group of some Lie group is a matrix Lie group.

As we know the universal covering group of $GL(n,\mathbb{R})$, $SL(n,\mathbb{R})$ is a Lie group which cannot be faithfully represented by a finite dimensional matrix. Therefore what's the sufficient ...
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72 views

Do I understand the Chevalley Restriction Theorem correctly?

Let $G$ be a complex semisimple Lie group with Lie algebra $\frak g$, and let $\frak h$ be a Cartan subalgebra with Weyl group $W$. The Chevalley Restriction Theorem states that the restriction map ...
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Show that $\left.\dfrac{d}{d s}\right|_{s=0}X_\eta (m)(f\circ \varphi_{\exp(s\xi )})=(X_\eta f)_{*,m}(X_\xi (m))$

Let $G$ be a Lie group, $M$ be a manifold, $\varphi:G\times M \rightarrow M$ smooth action of $G$ over $M$ where $\varphi(g,m)=g\cdot m$. For each $\xi \in \mathfrak{g}=T_e G$ we difine the ...