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

1
vote
1answer
43 views

If $ g \circ f$ is real analytic and $g$ is a real analytic immersion, then $f$ is real analytic

Let $M$ $N$ $P$ be complex manifolds, and let $$f: M\rightarrow N, g: N\rightarrow P$$ be $C^\infty$ maps with $g$ and $g\circ f$ holomorphic, and with $dg$ never degenerate. It's easy, then, to see ...
3
votes
2answers
30 views

How to descent to smaller groups “by chopping off a node of the Dynkin diagram”?

I read in section 2 of this paper : "There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram." What exactly is here referring to ...
1
vote
1answer
13 views

Why is the rank of a group is equivalent to the maximum number of independent U(1) generators?

I read here http://motls.blogspot.de/2012/04/exceptional-lie-groups.html that the rank of group is "the maximum number of independent U(1) generators". In my understanding the rank of a group ...
1
vote
0answers
12 views

Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
-1
votes
0answers
28 views

Proof theorem of Lie's Algebra [on hold]

I need help with a proof this theorems, anyone could be help? I need a proof this theorems. Corollary of Lie's Theorem: Let $L$ be a solvable subalgebra of $\text{gl}(V)$, $\dim V = n$ (finite). ...
0
votes
0answers
13 views

Dimension of adjoint orbits in $\mathfrak{su}(n)$

What is the dimension of the sub-manifold $M(A)$ of $\mathfrak{su}(n)$ defined by: $M(A) = \{U^{\dagger} A U \ \text{s.t.} \ U \in SU(n) \}$ for each $A \in \mathfrak{su}(n)$.
0
votes
0answers
15 views

Decomposition of an algebraic group into unipotent radical, component group and Levi subgroup

Consider the following statement : Given an algebraic group $G$ (over $\mathbb{R}$ or $\mathbb{C}$), there is a decomposition $$G = (G/G^o) \times \left( (G^o/R_u(G^o)) \ltimes R_u(G^o)\right)$$ ...
3
votes
1answer
30 views

Why not every homogeneous manifold is parallelizable?

It is obvious that not every homogeneous manifold is parallelizable (take for example the two-sphere $S^{2}$). In contrast, every Lie group $G$ is parallelizable, as you can construct a pointwise ...
1
vote
1answer
27 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can ...
4
votes
1answer
27 views

Representing Matrix Subgroups

Suppose I wanted to describe the subgroup of $GL_n(\mathbb{R})$ of matrices of the form $$ \left [ \begin{array}{cc} A & B \\ 0 & C \\ \end{array} \right ] $$ where $A \in ...
1
vote
0answers
19 views

Is trace of regular representation in Lie group a delta function? [duplicate]

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
2
votes
1answer
58 views

Is trace of regular representation in Lie group a delta function?

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
1
vote
1answer
24 views

Gauge transformation on a principal bundle

I am reading through lecture notes found here and on pg 11 they define a map $\overline{\phi}_{\alpha}:\pi^{-1}(U_{\alpha})\rightarrow G$ by ...
1
vote
1answer
54 views

direct sum and tensor product of representation of lie algebra

Let $(p_1,V_1)$ , $(p_2,V_2)$ representation of a lie algebra $g$ on $V_1,V_2$. I have to prove that: $ i) $ the direct sum $p_1 \oplus p_2$ is a representation of $g$ in $V_1 \oplus V_2$ $ ii) $ ...
0
votes
0answers
8 views

Commutation Relation - Generators of Semisimple Lie Group

The following is stated in a book I picked up, Group Theory for High-Energy Physicists - M. Saleem, M. Rafique. Consider an $r$-parameter semisimple Lie group of rank $\ell$. It has a set of ...
1
vote
0answers
29 views

Is there a general method to calculate the generators of the subgroups of $\textrm{GL}(n,F)$?

I know this might be a very bad/broad question, but after going through a few practice problems for finding linearly independent generators for some of the easier subgroups of $\textrm{GL}(n,F)$ ...
2
votes
1answer
38 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
31 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} ...
1
vote
0answers
23 views

$\partial_x + 2y\partial_z$, $\partial_y - 2x\partial_z$ exist, but $\partial_t$ doesn't

Let $X=\partial_x + 2y\partial_z$, $Y=\partial_y - 2x\partial_z$, and $Z=\partial_z$. If there is a function $f(x,y,z)$ and open set $U$ such that derivatives $Xf$ and $Yf$ exist on $U$, but $Zf$ ...
0
votes
0answers
24 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 ...
1
vote
0answers
21 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?
-1
votes
0answers
22 views

Lie groups and matrix groups

i am wondering the following question, are there Lie groups that are not isomorphic to a matrix group? Thanks, Emmanuel
0
votes
0answers
12 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 ...
3
votes
1answer
22 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
23 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
13 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
48 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
27 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
45 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
24 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
6 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
22 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
119 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
29 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
39 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
28 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
39 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
35 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
35 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
52 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
21 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
26 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
17 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
29 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
43 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
20 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
18 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
61 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
31 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 ...