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 ...

learn more… | top users | synonyms

2
votes
1answer
39 views

What it means SO* (2N)?

I'm puzzled about the $"*"$ in the following notation for Lie groups: $SO^* (2N)$ or $SU^* (2N)$. I don't understand what is the meaning of this notation. It is introduced for example in Gilmore ...
0
votes
1answer
36 views

Finite representations of the Euclidean Group

What are the finite dimensional indecomposable representations of the special Euclidean group in three-dimensions, SE(3)? To clarify, I'm asking about the group $$SE(3) = \left\{ \begin{pmatrix} ...
0
votes
0answers
33 views

Cartan's Criterion for Solvability

I'm trying to understand the proof of Cartan's Criterion for Solvability given here, and have two questions: On page 15, about half way down, we assert the following: If $\mathfrak{g}=\mathfrak{g}_0 ...
0
votes
0answers
22 views

How to write su(3) Lie algebra as a sum of two subspaces? [duplicate]

Let K,F⊂su(3) be subspaces, such that K⊕F=su(3), and K has a su(2) structure. How can we show that [K,K]=K (i.e., commutator of any two elements of K gives an element in K), [K,F]=F, and [F,F]=K?
0
votes
0answers
16 views

Adjoint representation for matrix groups (Gauge theory)

This is a question in regards to an identity in Gauge theory. Let $\omega$ be the connection one form on a principal bundle $\pi:P\rightarrow M$ and let $A_{\alpha}:=s_{\alpha}^*\omega$ be the gauge ...
4
votes
1answer
29 views

Sections of associated bundles

Let $\pi:P\rightarrow M$ be a Principal bundle and $\pi_V:P\times_G F\rightarrow M$ be its associated bundle via the representation $\rho:G\rightarrow GL(V)$. Fact: $\Gamma(P\times_G ...
0
votes
1answer
25 views

Connected Matrix Lie groups

I was reading Hall's book on Lie groups. After defining Connected Lie groups he stated and proved a proposition : If $G$ is a matrix Lie group then the component of $G$ containing identity is a ...
2
votes
0answers
18 views

Generators of the SU(2) matrix group

Let $X_i$ be the generators of $SU(2)$ and let the parameters of the rotation be $\theta, \phi, \delta$ such that the matrix $R = e^{i(\phi X_{1} + \delta X_{2} + \theta X_{3})}$, where $R$ is an ...
0
votes
0answers
50 views

Lie algebra of a lie group

In proposition 5.2 of Bump's Lie Groups, he states: Let $G$ be a closed Lie subgroup of $GL(n, \mathbb{C})$, and let $X \in Mat_n (\mathbb{C})$. Then the path $t \to exp(tX)$ is tangent to the ...
0
votes
1answer
30 views

The dimension of the SU(2) matrix group

Let's take the matrix $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Consider its transpose $R^\dagger = \begin{pmatrix} a^* & c^* \\ b^* & d^* \end{pmatrix}$. Then $RR^\dagger ...
0
votes
1answer
47 views

$\mathbb{R}^2$ as a quotient of a group

I am looking for a locally compact group $G$ with a closed subgroup $H$ such that $G/H$ is homeomorphic with $\mathbb{R}^2$ but $G$ does decompose into a semidirect and/or direct product which ...
0
votes
1answer
26 views

Dimensions of classical Lie groups

I understand that the dimension of the $SU(n)$ matrix group is $n^2$ because there are $2n^2$ real variables (for the $n^2$ complex matrix elements) in each matrix, and there are $n^2$ equations ...
0
votes
0answers
8 views

Question on Discrete series representations of semisiple Lie groups

I am reading Knapp's book, representation theory of semisiple Lie groups. I am confused with the statements in the following. In page 310, Theorem 9.20: Let $\lambda \in (i\mathfrak b)'$ is ...
0
votes
1answer
28 views

Property of left invariant vector field and its local flow.

Given $G$ a lie group and $X$ a left invariant vector field. Let $\Phi_X^t$ be the local flow of $X$. Why can we conclude that $\Phi_X^t \circ L_x=L_x \circ \Phi_X^t$? Thanks!
6
votes
1answer
130 views

Why are Lie algebras “rigid” objects?

I read the following motivation for quantum groups on wikipedia: The discovery of quantum groups was quite unexpected, since it was known for a long time that compact groups and semisimple Lie ...
0
votes
1answer
32 views

Lie Algebra associated to a lie group [closed]

Given an infinite dimension vector space, let $G=I+End^f(V)$ where $End^f(V)$ is the ideal of finite rank endomorphism, and $H=G_1\subset G$ of endomorphisme of determinant $1$. Could you help me ...
0
votes
1answer
42 views

Left invariant vector field

Let $G$ be a Lie Group with $e$ as the neutral element. Taken $X_e\in T_e G$, define $$X(a)=(dL_a)_e X_e$$ Why this vector field is left invariant? I get confused with the notation. Thanks!
0
votes
1answer
29 views

Group exponentials and general group of diffeomorphisms

I read on the wiki page (http://en.wikipedia.org/wiki/Exponential_map_%28Lie_theory%29) that the group exponential is not a local diffeomorphism at all points. Can someone give me an example?
1
vote
1answer
43 views

Which mathematical theory investigates distortions?

For transformations like rotations, translations or boosts Lie theory is the appropriate theory. Which theory talks in similar, systematic terms about distortions $$ M \vec{v} = m \vec{v} \quad ...
4
votes
0answers
37 views

What is the center of SO(p,q)?

For the matrix lie group $SO(p,q)$, what is in the center? Is there anything other than $I_n$ (or $-I_n$ in the case $p+q=n$ is even)? Also where can I find a reference?
0
votes
2answers
48 views

Semisimple implies complete reducibility

Why does a semisimple Lie algebra imply complete reducibility? I have that a semisimple Lie algebra is a Lie algebra with no non-zero solvable ideals. Complete reducibility means that every invariant ...
1
vote
0answers
53 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
0
votes
1answer
24 views

A doubt from the Isomorphism theorems of Lie algebras.

Given an isomorphism of two irreducible root systems $\Phi$ and $\Phi$' we need to show that the corresponding simple Lie algebras $L$ and $L'$ are isomorphic. For that we take the subalgebra $D$ of $ ...
0
votes
0answers
50 views

Adjoint of Exponential Map

If $\exp: T_p(G) \rightarrow G$ is the expoenential map of a lie group, then what does the adjoint operator (as in $\langle Ax,y\rangle=\langle x,A^*,y\rangle$) of the derivative of exp look like? ...
1
vote
0answers
18 views

Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...
3
votes
0answers
37 views

Second derivatives of rotations

Given an exponential parameterization of a 3D rigid rotation $R\in SO(3)$ by the vector $v = (v_x, v_y, v_z)^T$ I would like to find its second derivatives at the point $v=(0,0,0)$. Using the ...
1
vote
0answers
22 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
3
votes
1answer
46 views

What information can I immediately extract from a Dynkin diagram?

I have understood quite well how we construct Dynkin diagrams. My question is the following: What immediate information can I extract just by looking at a Dynkin diagram? Of course I can ...
0
votes
1answer
22 views

Metrics on affine connections

In one of the paper's I read this statement: "the affine geodesics of the Cartan connections (group geodesics) are metric-free". What does this really mean? paper: ...
0
votes
0answers
20 views

Why is the kernel of the connection one form a connection on a principal bundle?

Let $\pi:P\rightarrow M$ be a principal bundle and let $\omega\in \Omega(P;\mathfrak{g})$ be a one form satisfying $\omega(\sigma(X))=X$ and $R_g^*\omega=\text{Ad}_{g^{-1}}\circ\omega$ Then ...
0
votes
0answers
72 views

A question about a proof in Lang's $SL_2(\mathbb{R})$

The following is a lemma in Lang's book $SL_2(\mathbb{R})$. It's the last line of the proof that I don't understand. Let $G=SL_2(\mathbb{R})$ , $E$ a Banach space, and let $\pi$ be an irreducible ...
3
votes
1answer
37 views

İf x is diagonalizable then ad(x) is also diagonalizable

I start to study lie algebras from K. Erdmann, Mark J. Wildon-Introduction to Lie Algebras and i try to solve question below but actually i can't see .How can i start ? Give me hint please Let ...
4
votes
0answers
24 views

Isomorphism of Principal Bundles with structure groupoid

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Suppose that $\pi:P\rightarrow B$ is a $\mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)\rightarrow (P,B); s\in [0,1]$, be a homotopy ...
0
votes
1answer
30 views

Nested commutators that don't vanish

So I've been reading up on Lie Groups and Lie Algebras and the Baker-Campbell-Hausdorff formula. I understand how the formula works and that most of the time the nested commutators vanish at a certain ...
7
votes
1answer
90 views

$S^2=SO(3)/SO(2)$. Does this mean that $S^2 = SU(2)/U(1) $?

$S^2=SO(3)/SO(2)$. Does this mean that $S^2 = SU(2)/U(1) $ since $SO(3) \approx SU(2)$ and $SO(2) \approx U(1)$? Is there some more generic rule on how to relate $S^{n-1} = SO(n)/SO(n-1)$ to the ...
0
votes
1answer
35 views

Why for simple roots in Lie algebras the master formula reduces to one integer?

The master formula for two generic weights (roots) is $$ 2 \frac{\vec{a} \cdot \vec{b} }{\vec{a} \cdot \vec{a} }=q-p $$ but if we require that the roots are simple then this reduces to $$ 2 ...
1
vote
0answers
10 views

Inverting the the decomposition of tensor product representation into irreps

Suppose I have two unitary representations $U_V, U_W$ of a group $G$ on finite-dimensional vector spaces $V$ and $W$. I know that the tensor product representation $U_V\otimes U_W$ need not be ...
0
votes
1answer
41 views

Clarification on notation of “left invariant fields” (Lie groups)

In these notes in Definition 1.4 we learn that A vector field $X$ on a Lie group $G$ is called left invariant if $d(L_g)_h(X(h))=X(g(h))$ for all $g,h \in G$, or for short $(L_g)_*(X)=X$. where ...
2
votes
2answers
73 views

Normal Subgroups of $SU(n)$

I was wondering if there is any classification for normal subgroups of $SU(n)$? In particular, I think that the answer is no for $n = 2$ by looking at the covering map onto $SO(3)$, but I was curious ...
0
votes
0answers
46 views

What is the differential of left translation?

Let $G$ be a Lie group, $g\in G$ and $L_g$ be left translation by $g$. I want to compute the differential $dL_g|_0$ of $L_g$ at $0$. Attempt: Let $v\in T_0G$ be a tangent vector at $0$. Let ...
0
votes
1answer
30 views

Diffeomorphism in Lie Group

$G$ is a Lie group and consider $L_{g}: G \rightarrow G$ ($L_g(h)=gh$). What i need to show that $L_{g}$ is diffeomorphism. Is it something obvious? Can someone explain it to me?
0
votes
0answers
5 views

Proof that ideal in Lie ring can be represented as sum of 2 Lie subrings.

Let $K$ be a commutative ring and $m≥3$. Let $L(m,K)$ be a Lie subring of matrices with coefficients from ring $K$ that contains matrices with null traces, $L(m,K)={(a_{ij})∈M_m(K)|\sum\limits_{i = ...
6
votes
3answers
280 views

Lie group. How is manifold defined?

If I have the Lie group $SL(2,\mathbb{R})$. Then how is the manifold structure on this algebraic group defined, could anybody explain this to me? I mean this is the group of matrices that determinant ...
1
vote
1answer
26 views

Quaternions and Lie Groups

It's obvious that Quaternions, (denote by $H$, without $0$) form a non-commutative group under multiplication ( it's even non commutative division algebra ). It seems that it's also obvious that ...
1
vote
1answer
47 views

Are unipotent algebraic groups connected?

Is a unipotent algebraic group over a field of characteristic zero always connected?. As far as I know, every unipotent algebraic group over field of characteristic zero is isomorphic to a closed ...
0
votes
0answers
11 views

How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$?

How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$, where $ω = |ω|, ωˆ =ω/|ω|.$ My Attempt: In my understanding, $\pi (w)$ is an element of the lie algebra, which is a ...
0
votes
0answers
24 views

Why is left-invariant vector fields needed to construct a Lie algebra from a Lie group?

Since the set of all vector fields $V$ on a Lie group $G$ forms a vector space, one can impose algebraic structure (a Lie algebra) by defining the bracket $[\cdot,\cdot]$ between these vector fields. ...
2
votes
1answer
41 views

Diagonalizing elements of compact lie groups

Chapter 5 of Sepanski's Compact Lie Groups starts with this paragraph: "Since a compact Lie group $G$ can be thought of as a Lie subgroup of $U(n)$, it is possible to diagonalize each $g\in G$ using ...
2
votes
0answers
49 views

Any continuous group homomorphism from $\mathbb{R}$ to $GL(n,\mathbb{C})$

Any continuous group homomorphism $\phi$ from $\mathbb{R}$ to $GL(n,\mathbb{C})$ is of the form $\phi (t)=exp(tX)$ for some $X\in M(n,\mathbb{C})$. Can anyone give hints for the proof of this fact? I ...
1
vote
1answer
31 views

Exterior derivatives involving representations

I have two questions regarding the exterior derivative of vector valued forms when representations are involved: Suppose $V$ is a vector space, $M$ a smooth manifold and $\omega$ is a $V$ valued ...