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.

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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 $[\...
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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, $\pi : \mathfrak{g} \mapsto End(V)$)....
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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 ...
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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 ...
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  • 155
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1 answer
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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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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 ...
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  • 249
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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 ...
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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 ...
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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 ...
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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 $gl(V)$ ...
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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 ...
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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 $...
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3 votes
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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:\...
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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 ...
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2 votes
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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....
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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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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 ...
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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^{...
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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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 _\...
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2 votes
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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 ...
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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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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 $(M_n (\mathbb{F}) , [\cdot , \cdot ] )$ ...
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  • 1,558
1 vote
1 answer
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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 ...
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2 votes
1 answer
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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 ...
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1 vote
1 answer
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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 ...
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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
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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 $\mathfrak h$ ...
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3 votes
1 answer
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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 ...
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2 votes
1 answer
63 views

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 $\text{Ad}_X(y) = XyX^{-1}$. ...
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3 votes
1 answer
122 views

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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1 vote
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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 ...
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1 answer
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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}...
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3 votes
1 answer
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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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0 answers
39 views

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 ...
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  • 455
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0 answers
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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 ...
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1 vote
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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 ...
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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
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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 ...
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2 votes
1 answer
179 views

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 ...
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  • 1,212
1 vote
0 answers
50 views

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 $...
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  • 179
1 vote
1 answer
117 views

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 ...
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  • 1,212
4 votes
1 answer
116 views

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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