# 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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### 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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### 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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### 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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### 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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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 $X$, $P$, $X^{... 0 votes 0 answers 44 views ### Multiplicity of highest weights and representations I am wondering whether there is a nice and clean way of detecting the dimension of an irreducible representation of a complex simple Lie algebra simply from the linear expansion of the corresponding ... 1 vote 0 answers 15 views ### Longest element of Weyl group of a simple Lie algebra action on Weyl chambers Let,$\mathfrak{g}$be a complex simple Lie algebra with Weyl group$W$,also let$\omega_0$be the longest element of the Weyl group. We Know that Weyl group acts on the set of Weyl chambers freely ... 0 votes 1 answer 62 views ### Constant$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] = ... 0 votes 0 answers 34 views ### 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 ... 0 votes 0 answers 28 views ### 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 ... 3 votes 1 answer 35 views ### 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 ... 0 votes 0 answers 19 views ### 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 ... 0 votes 1 answer 44 views ### 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 _\... 2 votes 1 answer 95 views ### 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 ... 1 vote 0 answers 51 views ### 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 ... 1 vote 0 answers 35 views ### 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 vote 1 answer 37 views ### 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)$$My motivation for this question is this particular part from my representation theory lecture: We ... 2 votes 1 answer 86 views ### Claims on Root systems For a root system R prove or disprove: a. Assume that the angle θ between the roots α and β is obtuse (θ > π/2) Then α+β ∈R. b. The angle θ between α and β is π/2 . Then α+β is not a root. c. If ... 1 vote 1 answer 57 views ### What is the Lie theoretic interpretation of conjugate of a partition? For a partition \lambda it is very well-known operation to take its conjugate partition \lambda' which is obtained by transposing the Young diagram of \lambda. A partition \lambda can be ... 1 vote 0 answers 34 views ### Real forms of complex Lie algebras and Galois theory I have been reading about the classification of real simple Lie algebras, and one of the key theorems states: Let S be a semisimple complex Lie algebra. The map$$\Psi:\Bigg\{\begin{split}&\... 3 votes 1 answer 54 views ### 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$\mathfrak h$... 3 votes 1 answer 73 views ### 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 ... 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$\text{Ad}_X(y) = XyX^{-1}$. ... 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 ... 1 vote 0 answers 28 views ### 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 ... 0 votes 1 answer 70 views ###$e^XY$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 $$e^XY=\frac{\alpha}... 3 votes 1 answer 65 views ### Decomposition of trace zero real matrix with purely imaginary eigenvalues I would like to prove that for a matrix A\in sl(2,\mathbb{R}), i.e. a real matrix with \text{tr}(A)=0, if the eigenvalues of A are \pm i\alpha, for \alpha\in \mathbb{R}, then there is a real ... 0 votes 0 answers 39 views ### Derivations of algebraic independence I was wondering if this is this is true or not: If D is a derivation and x_1,x_2,...,x_n are algebraically independent, and p(x_1,...,x_n) is a homogeneous polynomial with all of its monomials ... 0 votes 0 answers 52 views ### Choosing a norm on a Lie algebra, that is "compatible" with the adjoint representation I have the following problem: Consider a compact Lie group G. Let \mathfrak{g} denote its Lie algebra. We have the exponential map:$$ exp: \mathfrak{g} \rightarrow G$$It is a fact that we may ... 1 vote 0 answers 48 views ### 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 ... 0 votes 1 answer 27 views ### 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? 0 votes 0 answers 30 views ### 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 ... 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 ... 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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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 ...
### 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 ...