Questions tagged [lie-algebras]
For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.
5,943
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$[\mathfrak{gl}(n,\mathbb{K}),\mathfrak{gl}(n,\mathbb{K})] = \mathfrak{sl}(n,\mathbb{K})$
Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$. How do I proove that: $[\mathfrak{gl}(n,\mathbb{K}),\mathfrak{gl}(n,\mathbb{K})] = \mathfrak{sl}(n,\mathbb{K})$?
I know that it is easy to see that $[\...
- 188
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16
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Prove that raising operator on non-highest weight is non-zero
Let $\mathfrak{h}$ denote the Cartan subalgebra of some Lie algebra $\mathfrak{g}$.
Let $(\pi,V)$ be a representation of the Lie algebra. ($V$ is the vector space, $\pi : \mathfrak{g} \mapsto End(V)$)....
- 11
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1
answer
40
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Is $[ad_X, ad_Y] = ad_X ad_Y - ad_Y ad_X$ the unique Lie bracket on $End(\mathfrak{g})$?
This follows from a similar question before(Adjoint map is a Lie homomorphism), but I have a question other than the statement itself.
I am trying to understand the proof of Proposition 3.8 in Brian ...
- 515
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35
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Algorithms to Compute Dimension of Lie Algebra
I am looking for fast algorithms to solve the following problem:
Let $\{b_1, b_2, \dots, b_m\}\subseteq M_{n}(\mathbb{C})$ be an independent set over $\mathbb{R}$. Find the dimension of the real lie ...
- 155
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1
answer
25
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Lie Algebra homomorphism from $\mathfrak{sl}(2,\mathbb{R})$ to $\mathfrak{gl}(m,\mathbb{R})$ is the derivtive of a unique Lie group homomorphism
I want to show that every lie algebra homomorphism $\phi$: $\mathfrak{sl}(2,\mathbb{R}) \rightarrow \mathfrak{gl}(m,\mathbb{R})$ is the derivtive of a unique Lie group homomorphism $\Phi: Sl(2,\mathbb{...
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35
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Definition of $Ad(r)$ the automorphism of a Lie-algebra $\mathfrak{g}$ of $G \subset SL(V)$.
Let $G \subset SL(V)$ be a connected algebraic group, acting irreducible on $V$. Consider the Lie-algebra $\mathfrak{g}$ of $G$, which is semisimple and acts irreducible on $V$.
Then i want to do the ...
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Showing that the Lie-algebra $\mathfrak{g}$ of $G \subset SL(V)$ is semisimple and irreducible on $V$. [solved]
Let $G \subset SL(V)$ be a connected algebraic group, acting irreducible on $V$, where $V$ is a complex vectorspace of dimension $n$
I want to show that the Lie algebra $\mathfrak{g}$ of $G$ is ...
- 249
1
vote
1
answer
32
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Zero element in Lie algebra of a Lie group
On page 189 of John Lee’s Introduction to Smooth Manifolds it is stated that the set of all smooth left-invariant vector fields on a Lie group $G$ is a linear subspace of the space of all smooth ...
- 45
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1
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26
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Recovering the definition of exponential matrix from the abstract definition of Lie groups.
I am studying the exponential function of the book introduction to the smooth manifold by John Lee and the following question has arisen.
Let $\exp:\mathcal{G}\to G$ exponential map, with $G$ a Lie ...
- 2,580
2
votes
2
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38
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Time derivative of the blend of a pair of quaternion curves
I have two curves ${\bf q}_0(t), {\bf q}_1(t)$. Each curve maps time $t$ to a unit quaternion. Construction of these curves is not important here, although we do have the respective time derivatives ...
- 51
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0
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23
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Proof of Lie's theorem [duplicate]
I am reading Humphrey's Introduction to Lie algebras and representation theory. I am stucked on the proof of theorem in Chapter 4.1 for this small point.
Assume $L$ is a solvable subalgebra of $gl(V)$ ...
- 135
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24
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What is the KAK (Cartan) decomposition in $\text{SL}(d, \mathbb R)$ in terms of linear algebra language (and its relation with SVD)?
Let $G= \text{SL}(d,\mathbb R)$ and consider its Cartan decomposition in the Lie group level $G=KAK$. Here $K$ should be the compact group $\text{SO}(d,\mathbb R)$ and $A$ is a diagonal matrix (please ...
- 41
4
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Opposite of the universal covering group
Let $\mathfrak g$ be a finite dimensional real Lie algebra and let $\tilde G$ be the unique simply connected Lie group with Lie algebra $\mathfrak g$.
I think that the set $R$ of connected Lie groups $...
- 197
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1
answer
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Which dimension is needed to represent a Lie algebra as a matrix algebra (as in Ado's Thm)?
Suppose $\mathfrak g$ is a real finite dimensional Lie algebra. If I understand it correctly, Ado's Theorem states that there is a real vector space $V$ and an injective Lie algebra homomorphism $\pi:\...
- 197
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Commuting generators of different Lie Algebras [closed]
Suppose that I have two different Lie algebras $\mathfrak{a}$ and $\mathfrak{b}$ with generators $M_{i}$ and $N_{j}$.
Is it always the case that, the commutator (which is assumed to exist throughout ...
- 99
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0
answers
39
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Tits' construction gives semisimple Lie algebras
Back to 1966, Jacques Tits gave a unified construction of the $5$ exceptional semisimple Lie algebras, and his work leads to the famous Freudenthal-Tits magic square. See, for example, https://arxiv....
- 157
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39
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Is the Maurer-form a 1-form?
I am studying Lie groups and lie algebras following Nakahara book. In 5.6.4 he introduces the concept of Maurer-Cartan one-form in this way:
\begin{equation}
\theta: X \rightarrow {(L_{g})^{-1}} _{*} ...
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0
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23
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Semisimple Lie algebra coincides with its commutant [duplicate]
I am trying to understand why for a semisimple Lie algebra $L$, we have $L = [L,L]$.
here are my thoughts.
Let $I:=[L,L]$ be a proper ideal in $L$, i.e., $I \ne L$, we then consider its orthogonal ...
1
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42
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On the structure of the set of Self-Adjoint operators acting on $L^{2}(\mathbb{R})$.
Let us consider the $Hilbert$ Space $L^{2}(\mathbb{R})$ and let $SA(L^{2}(\mathbb{R})$ be the space of all self adjoint operators acting on $L^{2}$. I have worked with operators such as $X$, $P$, $X^{...
- 116
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44
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Multiplicity of highest weights and representations
I am wondering whether there is a nice and clean way of detecting the dimension of an irreducible representation of a complex simple Lie algebra simply from the linear expansion of the corresponding ...
- 1,713
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15
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Longest element of Weyl group of a simple Lie algebra action on Weyl chambers
Let, $\mathfrak{g}$ be a complex simple Lie algebra with Weyl group $W$,also let $\omega_0$ be the longest element of the Weyl group. We Know that Weyl group acts on the set of Weyl chambers freely ...
- 11
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62
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Constant $SU(2)$ ASD connections on $\mathbb R^4$ are Flat
Let $A \in \mathfrak{su}(2) \otimes \mathbb R^4$, so $A$ is a collection of $4$ elements of $\mathfrak {su}(2), (A_0, \dots, A_3)$. We can consider the system of equations
$$
[A_0, A_1] + [A_2, A_3] = ...
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34
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Expanded Adjoint of se(3)
Follow up to this question about adjoint Matrix of se(3) Lie algebra: Adjoint of se(3)
I understand that the adjoint Matrix for Lie algebra is the derivative of the Lie group adjoint matrix, but does ...
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0
answers
28
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Understanding Lee J's Proposition 8.41
Hello. The image corresponds to Proposition 8.41 from the book introduction to smooth manifolds by Lee J. There are some statements in the proof that I can't understand at the moment. I put below what ...
- 2,580
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1
answer
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Are infinite-dimensional representations of semisimple Lie algebras semisimple?
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is well-known that $\mathfrak{g}$ is semisimple if and only if the category of finite-dimensional ...
- 493
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Proving that $\text{GL}(n,\mathbb{R})$ is open at $\text{gl}(n,\mathbb{R})$ and that its tangent space is isomorphic to $\text{gl}(n,\mathbb{R})$
Hello. The image corresponds to chapter 8 of vector fields of the book introduction to the smooth manifolds of the author Lee.
Question 1. Why $\text{GL}(n,\mathbb{R})$ is an open set of the vector ...
- 2,580
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1
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44
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The complexification of a compact connected Lie group, is it semi-simple?
I have read that the complexification of a compact Lie group is a reductive Lie group.
Question: Let $G$ be a compact connected Lie group and let $G _\mathbb{C}$ be a complexification of $G$. Is $G _\...
- 664
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1
answer
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Possible lenghts in an irreducible root system
If a root system $R$ is irreducible (not a product of two root systems) then $R$ does not contain three vectors of pairwise different lengths.
To show this do we need just to compute all the angles ...
- 820
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answers
51
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Extension of an involution on $G$ to an involution on $G_\mathbb{C}$
Let $G$ be a compact connected Lie group and $ \sigma :G \rightarrow G $ be an involution on $G$. Let $G^\sigma :=\lbrace g \in G, \sigma(g)=g \rbrace$.
Denote by $G_\mathbb{C}$ the complexification ...
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0
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35
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Lie algebra of general linear group
I’m reading through some Lie algebra notes and I have never studied Lie groups at all but it says that $\mathfrak{gl}_n (\mathbb{F})$ denotes the Lie algebra $(M_n (\mathbb{F}) , [\cdot , \cdot ] )$ ...
- 1,558
1
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1
answer
37
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Adjoint representation as derivative of adjoint map
Question: Let $G$ be a Lie group, how can we associate
$$T_e(\text{Aut}(T_eG)) = \text{Der}(T_eG)$$
My motivation for this question is this particular part from my representation theory lecture:
We ...
2
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1
answer
86
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Claims on Root systems
For a root system R prove or disprove:
a. Assume that the angle θ between the roots α and β is obtuse (θ > π/2) Then α+β ∈R.
b. The angle θ between α and β is π/2 . Then α+β is not a root.
c. If ...
- 820
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1
answer
57
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What is the Lie theoretic interpretation of conjugate of a partition?
For a partition $\lambda$ it is very well-known operation to take its conjugate partition $\lambda'$ which is obtained by transposing the Young diagram of $\lambda$.
A partition $\lambda$ can be ...
- 207
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0
answers
34
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Real forms of complex Lie algebras and Galois theory
I have been reading about the classification of real simple Lie algebras, and one of the key theorems states:
Let $S$ be a semisimple complex Lie algebra. The map $$\Psi:\Bigg\{\begin{split}&\...
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1
answer
54
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Why are highest weight modules of integral highest weight $B$-equivariant?
Suppose $G$ is a connected semisimple algebraic group over $\mathbb{C}$, $B \subset G$ is a Borel subgroup, and $T \subset B$ is a maximal torus. Write $\mathfrak g$, $\mathfrak b$ and $\mathfrak h$ ...
- 713
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1
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An explicit description of all conjugations on $\mathbb{C}$
The "standard" conjugation for a complex number $a+bi$ is $\overline{(a+bi)}=a-bi$.
If we see $\mathbb{C}$ as a one dimensional abelian Lie algebra, we can associate to this conjugation a ...
- 389
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1
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Adjoint map $\text{Ad}: G \to GL(\mathfrak{g}), X \mapsto \text{Ad}_X$ is continuous
Let $G$ be a matrix Lie group and $\mathfrak{g}$ the associated Lie algebra. For fixed $X \in G$, define the linear map $\text{Ad}_X : \mathfrak{g} \to \mathfrak{g}$ by $\text{Ad}_X(y) = XyX^{-1}$. ...
3
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What is the importance of Cartan decomposition of a semi-simple Lie algebra?
I just started learning about Cartan decomposition of semi-simple Lie algebras, and I'm curious to know what are their applications in studying semi-simple Lie algebras. My guess was that it might be ...
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Admissibility necessary for a $(\mathfrak{g}, \mathfrak{k})$-module to be a direct sum of simple $\mathfrak{k}$-modules?
I am now looking over the book A. Borel, N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Volume 67 of Mathematical Surveys and Monographs, AMS. I would ...
- 370
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70
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$e^XY$ and $Ye^X$
Let $X,Y$ be two matrices, and we define
$$
e^X:=\sum_{k=0}^{\infty}\frac{1}{k!}X^k
$$
In a problem about Lie algebras, I need to show if $[X,Y]=\alpha Y,\alpha\neq 2\pi ik$, then
$$
e^XY=\frac{\alpha}...
- 697
3
votes
1
answer
65
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Decomposition of trace zero real matrix with purely imaginary eigenvalues
I would like to prove that for a matrix $A\in sl(2,\mathbb{R})$, i.e. a real matrix with $\text{tr}(A)=0$, if the eigenvalues of $A$ are $\pm i\alpha$, for $\alpha\in \mathbb{R}$, then there is a real ...
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Derivations of algebraic independence
I was wondering if this is this is true or not:
If $D$ is a derivation and $x_1,x_2,...,x_n$ are algebraically independent, and $p(x_1,...,x_n)$ is a homogeneous polynomial with all of its monomials ...
- 455
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Choosing a norm on a Lie algebra, that is "compatible" with the adjoint representation
I have the following problem:
Consider a compact Lie group $G$. Let $\mathfrak{g}$ denote its Lie algebra. We have the exponential map:
$$
exp: \mathfrak{g} \rightarrow G
$$
It is a fact that we may ...
1
vote
0
answers
48
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Semisimple 6-dimensional Lie Algebra
I am working on Exercise 10.7 of Erdmann & Wildon's "Introduction to Lie Algebras":
Suppose $L$ is semisimple of dimension $6$. Let $H$ be a Cartan subalgebra of $L$ and let $\Phi$ be ...
- 716
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1
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27
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Is $\mathfrak{sl}(n,k)$ solvable/nilpotent?
I know $\mathfrak{sl}(n,k)$ isn't solvable if char$k\ne2$, and also $\mathfrak{sl}(2,k)$ is nilpotent if char$k=2$. What about $\mathfrak{sl}(n,k)$ when char$k=2$ in general?
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0
answers
30
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Prove an endomorphism is nilpotent [duplicate]
Let $V$ a finite dimensional vector space over a characteristic zero algebraically closed field. Let $x,y\in\mathfrak{gl}(V)$ and $[x,y]=z$ such that $z$ commutes with $x$ and $y$. Prove $z$ is a ...
- 43
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1
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179
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Which Dynkin diagram is being spoken about here? Why is there a double line?
I'm confused about the following comment in Knapp's Lie Groups 2ed, page 397. Here, $\Delta$ is a root system associated to a complex semisimple Lie algebra, $\alpha, \beta$ are orthogonal roots and ...
- 1,212
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0
answers
50
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Do there exist nontrivial bi-invariant differential forms on $SL_2(\mathbb{R})$? [closed]
More specifically, $1$-forms and/or $2$-forms.
Since $SL_2(\mathbb{R})$ is not compact the usual averaging trick doesn't work.
If the answer is no, do there exist nontrivial Ad-invariant forms on $...
- 179
1
vote
1
answer
117
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What is a simple component of a root system?
The above is from Knapp's Lie Groups; Beyond an introduction', 2ed, page 397.
Question 1: What is a simple component of a reduced root system? Or, more specifically, given a root system $\Delta$ and ...
- 1,212
4
votes
1
answer
116
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The eigenvalues of the generators of $SO(N)$
What are the eigenvalues of generators in the irreducible representations of the Lie algebra of $SO(N)$ (and its doublecover)?
For me, the naive generators of $SO(N)$'s fundamental representation are ...
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