All Questions
Tagged with general-linear-group lie-groups
30 questions
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$U_2$ homeomorphic to $S^3 \times S^1$, how? [duplicate]
Recently, I have been reading Linear Groups, chapter 9, from Michael Artin, Algebra book.
There is a question in the exercise (Chapter 9, Exercise section 3, no.3.2), given as follows
Prove that $U_2$ ...
- 51
0
votes
1
answer
44
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Understanding the Terminology of Closed Subsets in $GL(n, \mathbb{C})$
In the context of the general linear group $GL(n, \mathbb{C})$, we use specific terminology to describe closed subsets. I'm paying attention to two definitions, which I report here and which come from ...
- 359
-1
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1
answer
60
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Quotient group structure of general linear group
I want to find all quotient group structure in $GL(\mathbb{R}^n)$, which is all n*n invertible matrix on real number. The quotient group structure in $GL(\mathbb{R}^n)$ means to find normal subgroup $...
- 94
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1
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110
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When is the quotient of $\operatorname{GL}_n(\mathbb{R})$ by a discrete subgroup compact?
My question is exactly that on the title.
I'm interested in the action of some (discrete) subgroup $H$ on $\operatorname{GL}_n(\mathbb{R})$ by left multiplication.
For example, $H$ can be
$\...
- 18.7k
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0
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92
views
Show that as groups, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$, with the isomorphism given by $\theta \mapsto A_\theta$
Show that as groups, $\mathrm{GL}(V)\cong\mathrm{GL}_n(\mathbb{F})$, with the isomorphism given by $\theta \mapsto A_\theta$. The notation can be found in my attempt.
1st Attempt:
If $\dim_\mathbb{F} ...
11
votes
1
answer
833
views
What are the continuous outer automorphisms of the general linear group?
Is the only continuous outer automorphism of $\operatorname{GL}(n, \mathbb{R})$ the transpose inverse map $g \mapsto (g^\intercal)^{-1}$? If not, what other continuous outer automorphisms are there?
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1
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184
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Showing that the isomorphism of the general linear group of a vector space with the group of invertible matrices is smooth
This is Example 7.3(e) from John Lee's Introduction to Smooth Manifolds.
If $V$ is any real or complex vector space, $GL(V)$ denotes the set of invertible linear maps from $V$ to itself. It is a group ...
- 5,891
1
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1
answer
59
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Proof that $GL(n,\mathbb{C})$ is isomorphic to a properly embedded Lie subgroup of $GL(2n,\mathbb{R})$
Below is an example of embedded Lie Subgroup from John Lee's Introduction to Smooth Manifolds example 7.18 (d).
In this example, why is the image of $\beta$ a properly embedded Lie subgroup of $GL(2n,\...
- 5,891
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1
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49
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Query Regarding the Proof of Z(SO(2m))={±1} in Section 3.7 of 'Naive Lie Theory' by John Stillwell
I have been reading John Stillwell's 'Naive Lie Theory' and in Section 3.7, the author tries to prove that $Z(SO(2m)) = {\pm 1}$. In the proof, a matrix $I^\star$ is introduced with the form:
$$ I^\...
- 159
0
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0
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42
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Representations of general linear Lie algebra vs general linear group [duplicate]
I know that $\mathrm{GL}_{n}(\mathbb{C})$ is not simply connected. Therefore I don’t quite understand the correspondence between representations of $\mathrm{GL}_{n}(\mathbb{C})$ and $\mathfrak{gl}_{n}(...
- 581
1
vote
1
answer
107
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Why $SL$-invariants and highest weight vectors of rectangular shape coincide?
The groups $\mathrm{SL}(n) := \mathrm{SL}(n,\mathbb{C})$ and $\mathrm{GL}(n) = \mathrm{GL}(n,\mathbb{C})$ acts on $\mathbb{C}^n$ by multiplication from the left. This induces the diagonal action on $\...
- 125
3
votes
1
answer
143
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Is $GL(n)\otimes GL(n)$ is closed in $GL(n^2)$?
Let $GL(n) := \mathrm{GL}(n, \mathbb{C})$ - be space of invertible $n\times n$ matrices over $\mathbb{C}$, i.e. matrix Lie group.
Let $H := GL(n) \otimes GL(n)$ be a group with multiplication given by ...
- 125
2
votes
1
answer
253
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$GL_n(C)$ is isomorphic to a lie subgroup of $GL(2n,R)$
$GL_n(C)$ is isomorphic to a lie subgroup of $GL_{2n}(R)$.
I see some posts concerning this (not the same claim):
$GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
$GL_n\mathbb{C}...
- 567
0
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0
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122
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Chapter 1, Exercise 12 in Brian Hall's Lie groups, Lie algebras, and representations
Suppose $A$ and $B$ are invertible $n \times n$ matrices. Show that there are only finitely many complex
numbers $\lambda$ for which $\text{det}( \lambda A + (1-\lambda)B) = 0$. Show that there exists ...
- 61
2
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1
answer
1k
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Show that the general linear group $GL(n, \Bbb R)$ is a Lie group under matrix multiplication.
Show that the general linear group $GL(n, \Bbb R)$ is a Lie group under matrix multiplication.
I'm reading an introduction to manifolds by Tu and found this problem there. The definition I have for a ...
- 105
2
votes
1
answer
120
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Are there holes in every neighbourhood in GL(n,R)?
$GL(n,\mathbb{R})$ can be seen as $\mathbb{R}^{n^2}$ with points of non invertible matrix removed which form "holes" in $\mathbb{R}^{n^2}$. These holes seems to be dense in $\mathbb{R}^{n^2}$...
- 533
1
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162
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A question on the conclusion of the proof of "$Sl(2, \Bbb R)$ is transitive on the upper half plane".
$Sl(2, \Bbb R)$ is transitive on the upper half-plane. Here I have a question on the conclusion.
I have proved that for any $w \in \Bbb H$ there exists $z \in \Bbb H$ s.t $A⋅z=w$ for some $A \in Sl(2, ...
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-2
votes
1
answer
333
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$SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3,\mathbb{R})?$ or $SL(2,\mathbb{C})\cong \operatorname{Spin}(3,1,\mathbb{R})?$ [closed]
Which one of the isomorphism is correct?
$$SL(2,\mathbb{C})\cong \operatorname{Spin}(1,3,\mathbb{R})?$$
$$SL(2,\mathbb{C})\cong \operatorname{Spin}(3,1,\mathbb{R})?$$
If $SL(2,\mathbb{C})\cong \...
- 3,787
3
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0
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162
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A specific coset decomposition of $\mathrm{GL}_n(\mathbb{C})$
I am reading an old paper [1] where they introduce a specific decomposition of a complex, invertible matrix. There is no proof, so I am trying to come up with one. The claim goes as follows:
Let $\...
- 31
2
votes
1
answer
565
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Closed subgroups of $GL(n, \mathbb{C})$
Let $M_{n}(\mathbb{C})$ be the space of all complex $n\times n$ matrices equipped with the operator norm:
$$||A|| := \sup_{||x||\le 1}||Ax|| \quad A \in M_{n}(\mathbb{C})$$
Let $GL(n, \mathbb{C})$ be ...
- 4,085
0
votes
1
answer
289
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$GL_n\mathbb{C}$ as subgroup of $GL_{2n}\mathbb{R}$
I am confused about the following paragraph in Fulton W., Harris J. "Representation Theory: A First Course":
Of course, the group $GL_n\mathbb{C}$ of complex linear automorphisms
of a ...
- 35
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1
answer
215
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about Cartan's theorem
theorem : Let $SL(n,\Bbb{C})$ be the group of matrices of complex entries and determinant $1$ then $SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$.
proof : $GL(n,\mathbb{C})$ is a Lie ...
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-1
votes
1
answer
305
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How to determine the order of an element of Special Linear Group
Let,
$S=\begin{pmatrix} 0 &-1 \\ 1 & 0 \end{pmatrix}$
and
$T=\begin{pmatrix} 1 &1 \\ 0 & 1 \end{pmatrix}$ .
The matrices $S$ and $T$ generate $SL_2(\mathbb Z)$. The matrix $...
3
votes
2
answers
2k
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Generators for general / special linear groups
How do we determine the generators and relations for general / special linear groups over a finite field? I will be particularly interested in prime fields, in that also for $\mathbb{Z}_2$?
I know ...
- 7,165
1
vote
1
answer
78
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How to show that $\begin{pmatrix} 0 & -x \\\ 1/x & 0\end{pmatrix}$ is conjugate to a rotation?
Let $x >0$, and set $A=\begin{pmatrix} 0 & -x \\\ 1/x & 0\end{pmatrix}$.
Question: How to show that $A \in \operatorname{SL}_2(\mathbb R)$ is conjugate to an element of $\operatorname{SO}(2)...
- 25.6k
4
votes
1
answer
291
views
$PSl_n(\mathbb{C})\cong PGl_n(\mathbb{C})$?
I was reading about projective linear groups because I was asked to show that $PSl_n(\mathbb{C})\cong PGl_n(\mathbb{C})$. Here $PSl_n(\mathbb{C})$ is the projective space of $Sl_n(\mathbb{C})$, i.e. $...
- 4,336
0
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1
answer
132
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Continuity of a Group Homomorphism, Representation of SU(2)
In page 82 of Brian C. Hall's Lie Groups, Lie Algebras and Representations, he establishes the following representation $\Pi_m$ of SU(2) on $V_m$, the space of homogeneous polynomials of degree $m$ in ...
- 905
0
votes
2
answers
71
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Subspace Topology of $GL_n(\Bbb R)$
I'm trying to understand $GL_n(\mathbb R)$. I've read that we can think its topology with Euclidean metric. I know that $GL_n(\mathbb R)$ is a subspace of $Mat_n(\mathbb R)$ but I cannot understand ...
- 1,313
6
votes
1
answer
220
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Which linear maps $\mathbb{R}^{n^2} \to \mathbb{R}^{n^2} $ map $\text{GL}_n$ into $\text{GL}_n$?
Can we characterize all linear maps $\mathbb{R}^{n^2} \to \mathbb{R}^{n^2} $ which map $\text{GL}_n$ into $\text{GL}_n$?
In particular, is it true that every such map is given by
$X \to AXB$ or $X \...
- 25.6k
9
votes
2
answers
1k
views
Irreducible finite dimensional complex representation of $GL_2(\Bbb C)$
I know the basic theory of representation theory of compact Lie groups and I want to understand a non-compact example:
How to find all irreducible finite dimensional complex representations of $...
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