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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18 views

Deriving the commutation relations of the so(n) Lie algebra

The generators $(A_{ab})_{st}$ of the $so(n)$ Lie algebra are given by: $$(A_{ab})_{st} = -i(\delta_{as}\delta_{bt}-\delta_{at}\delta_{bs}) = -i\delta_{s[a}\delta_{b]t}$$, where $a,b$ label the ...
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20 views

Dimension of SO(n) and its generators

The generators of $SO(n)$ are pure imaginary antisymmetric $n \times n$ matrices. How can this fact be used to show that the dimension of $SO(n)$ is $\frac{n(n-1)}{2}$? I know that an antisymmetric ...
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39 views

Infinite-dimensional Unitary representions that are not completely reducible

The Peter-Weyl theorem asserts that for a compact Lie group $G$ every unitary irreducible representation is necessarily finite-dimensional and any unitary representation is a direct sum of ...
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16 views

Can an analytic function defined on a maximal torus be extended analytically to all the Lie group?

Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?
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27 views

Is the maximal torus a conjugacy class?

Let $G$ be a compact Lie group and consider $T$ a maximal torus in $G$. At Wikipedia I've read that $T$ is a conjugacy class of subgroups of $G$. Does it means that there exist $t \in G$ such that $T ...
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95 views

The second integral of the Killing form

Let $G$ be a lie group. Assume that $B$ is the Killing form of its Lie algebra $T_{e}G$. So $B$ is counted as a symmetric $2$-form on $G$ by translation. Is there a smooth function $f$ on $G$ ...
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14 views

Questions about vector fields on the upper half plane.

I am reading the lecture notes. On page 46, why $R_{X}$ as a vector field on $\mathcal{H}$ is $L_{pXp^{-1}}X$? Why $R_{\kappa} = 0$, $R_{\alpha}=2y \frac{\partial }{\partial y}$, ...
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28 views

Left invariant metrics on a Lie group coming from Lie algebras

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. If we have an inner product on $\mathfrak{g}$ then we can create a left invariant metric $d_G$ on $G$ by translations. On the other hand ...
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33 views

$\mathrm{ad}$-invariant splitting of Lie algebras.

I am reading a text on which the following situation appears: Let $G$ be a a Lie group and $H$ a compact Lie subgroup. Let $\mathfrak{g}$ and $\mathfrak{h}$ be their Lie algebras respectively. The ...
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29 views

Lie algebra of the invertible morphisms of a Lie algebra.

I am confused with some facts of Lie groups and Lie algrebras. If a have a Lie algebra $\mathfrak{g}$ and take its set of invertible morphisms $GL(\mathfrak{g})$ it is clear to me that this is a ...
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42 views

How to prove this sum being a real number related to unitary matrices?

Based on my recent study on non-Abelian gauge theory in physics, I encounter an identity that should be correct physically but I don't know how to prove it mathematically. Consider a $n\times n$ ...
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19 views

Hermitian metric from Killing form

Let $G$ be a semisimple Lie group. Its Killing form is a nondegenerate inner product on the tangent space to $G$ at the identity, and this form can be naturally extended to a metric on the whole of ...
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25 views

The Levi-Civita connection on $S^3$ and $SU(2)$

The fundamental theorem of Riemannian geometry implies that there is a unique symmetric (i.e., $\Gamma^{a}_{bc}=\Gamma^{a}_{cb}$, using a coordinate basis) connection on the three-sphere, $S^3$ which ...
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1answer
26 views

When is $SO(m,n)$ simple as a Lie group? What are the Zariski and Euclidean components?

Let $SO(m,n)=\operatorname{SO}(m,n)(\mathbb{R})$ denote the real $(m+n) \times (m+n)$ matrices, with determinant $1$, which preserve the quadratic form $x_1 + \cdots + x_m - x_{m+1} \cdots - x_{m+n}$ ...
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1answer
48 views

Infinitesimal Generator of Local Group of Transformation

I am stuck on concept of infinitesimal generator. I am reading Olver and i quote definitions from there Given a local group of transformation G acting on Manifold M via $g.x= \Psi(g,x)$ for $(g,x)\in ...
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60 views

Where have we used that $G$ is a compact Lie group?

Let $G$ be a compact Lie group acting on a topological space $X$. I need to show that the orbit map $p:X\rightarrow X/G$ has the path lifting property. Here is the proof - Let ...
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175 views

Visualization of SU(3)

I am trying to visualize the $SU(3)$ group used in quantum field theory. I have a (reasonably) good understanding of $SU(2)$ as the double cover of $SO(3)$ and also that this is homeomorphic to $S^3$. ...
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19 views

Analogies between finite groups and Lie groups

I believe there are some striking analogous facts between finite groups and Lie groups. One analogue almost too basic to mention is the appropriate notion of subobjects. In elementary group theory ...
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46 views

Sum of traces over Weyl group

I'm interested in computing sums like $\sum_{\sigma \in W} tr(\sigma ^3)$ , where $W$ is the Weyl group of $SO(2n+1)$, i.e. $W = (\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$. I tried to figure out what an ...
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1answer
23 views

Is there an analogue/extension of Baker-Campbell-Hausdorff formula for the conjugate?

Let $G$ be a Lie group (I only care about finite dimensional connected simply connected nilpotent groups, if that makes the answer easier). Let $\mathfrak g$ be its Lie algebra and let ...
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13 views

Index of Hessian of Composition of maps

Given a Lie group $G$ and a pair of smooth maps: $f:\mathbb{R}^n \rightarrow G \\ g: G \rightarrow [0,1]$ with $g$ possessing a single global optima, but potentially many saddle points. consider ...
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26 views

Immersed subgroup of a Lie group is a Lie group?

Let $G$ be a Lie group, $H$ a subgroup of $G$, which is an immersed submanifold of $G$. Kirillov, in his book "Introduction to Lie Groups and Lie Algebras" claims* it's easy to see $H$ will be a Lie ...
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21 views

De Rham Differential for Vector Valued Forms?

Let $M$ be a smooth manifold of dimension $n$ and let $V$ be a $\mathbb R$-vector space of finite dimension $\ell$. A $k$-form on $M$ with values on $V$ is a map $\omega$ on $M$ such that: ...
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22 views

An apparent contradiction in the simple Lie algebra $E_8$

The following is the Dynkin diagram for simple Lie algebra $E_8$ My question is the following: It is clear that $e_i+e_j$ for $i \neq j$ is a positive root. Let $\alpha _8$ be the fundamental ...
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19 views

Lie Symmetries: How to check b.c. of ODE without determining explicit symmetry solution?

I am trying to solve this ODE using symmetry methods. $$y''(x)=ay'(x)$$ With the boundary condition $y(0)=0$ and $y(\infty)=y_\infty $. The base solution is $y(x)=y_{\infty}(1-e^{ax})$. I found 8 ...
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1answer
37 views

Find parametrization for a possible “ruled” surface in $\mathbb R^4$

Let us endow $\mathbb R^4$ with a group law $\cdot$ such that the dilations $\delta_\lambda:(\mathbb R^4,\cdot)\to (\mathbb R^4,\cdot), (x_1,x_2,x_3,x_4)\mapsto (\lambda x_1,\lambda x_2,\lambda^2 ...
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45 views

Maximizing the pairwise Frobenuis distance between M othrogonal matrices

I want to maximize the pairwise Frobenius distance between $M$ orthogonal matrices. That is, I'm looking for $Q_{i}, i = 1, 2, ... M$ such that \begin{equation*} \begin{aligned} & \underset{ 1 ...
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60 views

How to calculate the Lie algebra of a neural network?

Define $F$ as the standard multi-layer feed-forward perceptron: \begin{equation} F(\mathbf{x}) = \Theta( W_1 \circ \Theta( W_2 \circ .... W_L(\mathbf{x}))) \end{equation} where $\Theta$ is the sigmoid ...
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1answer
32 views

$GL(n, \mathbb{C})$ is a properly embedded Lie subgroup of $GL(2n, \mathbb{R})$

Let $n$ be a positive integer, and define a map $\beta : GL(n, \mathbb{C}) \rightarrow GL(2n, \mathbb{R})$ by $$ \beta \begin{pmatrix} a^1_1 + i b^1_1 & \cdots & a^n_1 + i b^n_1\\ \vdots ...
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47 views

What background is needed to self-study/learn lie group? Which do books recommend? [closed]

What background is needed to self-study/learn lie group? Which do books recommend?
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19 views

Symmetry of Kahler metric on based loop group

The based loop group, $\Omega G$, is known to admit a Kaehler metric, given as \begin{equation} g(X,Y)=2\sum_{k>0}k\textrm{Tr}(X_{-k}Y_k), \end{equation} this is given in page 150 of Segal and ...
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1answer
41 views

Commutativity of a Lie algebra $\Rightarrow$ the Lie group is abelian

Let $G$ be a Lie group, $\mathfrak{g}$ it's Lie algebra. Assume $[x,y]=0 \, \, \forall x,y \in \mathfrak{g}$. Is it true that $G$ is abelian? Remarks: (1) The other direction ($G$ abelian ...
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26 views

Jacobian of Rotation composition

I need to compute Jacobians from compositions of rotations. E.g. $R = R_1 R_0$ \begin{align} \frac{\partial R}{\partial R_0} = ?\\ \frac{\partial R}{\partial R_1} = ?\\ \frac{\partial R_1 ...
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14 views

Solving Lie's invariant condtion for first order ODE

I write down the general equation $Y_{x}+(Y_{y}-X_{x})F-X_{y}F^{2}= XF_{x}+YF_{y}$ and assume that X=a(x) and Y=b(x)y, after that I can't see anyway to solve it for a and b. How can I get solution ...
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1answer
31 views

Exactness of Lie algebra exact sequence

If $G\rightarrow H\rightarrow K$ is an exact sequence of Lie groups, then I want to show that the induced sequence $\mathfrak{g}\rightarrow\mathfrak{h}\rightarrow\mathfrak{k}$ in Lie algebras is ...
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2answers
42 views

Prove that two Lie algebras are equal

Suppose to have a Lie group $\mathbb G$ whose Lie algebra $g$ admits a stratification $g=V_1\bigoplus V_2$ with $\text{dim} V_1=m$ and $g=Lie(V_1)$, i.e. $g$ is generated by $V_1$. In other words, ...
2
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1answer
51 views

Relation 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 usual Lie algebra of vector fields on $M$; that is ...
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1answer
20 views

Inner Product of Formal Lie Algebra Characters?

The inner product of characters of a compact Lie group is defined by integrating over the group. Question: Is it also possible to compute the inner product combinatorially from the formal characters? ...
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40 views

Outer automorphism group of $\mathfrak{su}(n)$

I'm reading about the special unitary Lie algebras, and seen it said that complex conjugation is not an inner automorphism of $\mathfrak{su}(n)$ for $n>2$. If there an easy way to see this? I ...
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27 views

Intersection of a family of closed Lie subgroups

If I have a Lie group $G$, and $\{H_{\alpha}\}_{\alpha\in A}$ is a family of closed Lie subgroups with Lie algebra $\{\mathfrak{h}_{\alpha}\}_{\alpha\in A}$, it's easy to see that $\bigcap_{\alpha} ...
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1answer
17 views

Exponential of adjoint action notation confusion

I'm getting a bit confused about the notation with the exponential map and the adjoint action. Could someone explain to me what $$ \text{exp}(t\text{ } ad(X))(Y) $$ means, where $X,Y$ are both ...
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2answers
58 views

Isotropic left invariant Riemannian metric on $GL_n^+$?

I am trying to see if it's possible to construct a left invariant isotropic Riemannian metric on $GL_n^+$. (the group of $n \times n$ invertible real matrices with positive determinant) (When by ...
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46 views

CFT's vs Vertex Operator Alagebras

I am trying to clear my ideas about the relation between a Conformal Field Theory (CFT) and a Vertex Operator Algebra (VOA). For me a CFT based on a (complex) vector space $H$ is a projective monoidal ...
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1answer
103 views

If a complex Lie group has the structure of an algebraic group, is this structure unique?

If $G$ and $H$ are algebraic groups over $\mathbb{C}$, and $f : G \rightarrow H$ is an isomorphism of complex Lie groups (i.e. a biholomorphic group isomorphism), then must $f$ be algebraic? If not, ...
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40 views

Action of discrete subgroup of Lie Groups

Given $\Gamma$ a discrete subgroup of a lie group $G$ I want to show that the action is wandering: $\forall x\in G\exists U_x\vert \{\gamma\in\Gamma\vert \gamma U_x\cap U_x\neq \emptyset\}$ is finite ...
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23 views

The Taylor series for product of Lie group elements

Let $x$ and $y$ be two elements of a Lie group $G$. In chapter 2 of the text "Lie Groups and Lie Algebras I" by A. L. Onishchik, the author states that, if $\overline{x}$ and $\overline{y}$ denote the ...
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71 views

Isoclinic rotations in four dimensions

Given any collection of complementary, oriented (2D) planes in n-dimensional space, and an angle associated with each one, there is a unique rotation of the whole space which restricts to rotations in ...
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1answer
206 views

Orbits of $SL(3, \mathbb{C})/B$

Let $B= \Bigg\{\begin{bmatrix} * & *&* \\ 0 & *&*\\ 0&0&* \end{bmatrix} \Bigg\}< SL(3,\mathbb C)$. What is $SL(3,\mathbb C)/B$? Do we use these facts: Borel fixed ...
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1answer
17 views

Question about Bruhat decomposition for $SL_2$.

I am reading the lecture notes. On page 11, let $G=SL_2$. It is said that $G = NwP \cup P = PwN \cup PwNw^{−1}$, where $P= \left\{ \left( \begin{matrix} a & x \\ 0 & 1/a \end{matrix} \right) ...
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1answer
27 views

How to show that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures?

I am reading the lecture notes. On page 19, let $G=SL_2$. It is said that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures on the ...