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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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The action map induces a Lie algebra homomorphism

Let $G$ be a Lie group and $M$ be a manifold such that $G$ acts on $M$ transitively. Let $$\phi: G\times M\to M$$ be the action map. Now $\phi $ induces a homomorphism $$\alpha:G\to Aut(M)$$ How to ...
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Problem with universal enveloping algebra generated by a single element (Jacobson)

Jacobson's book on "Lie algebras" has the following definition of enveloping algebra generated by a subset (Definition 2, Chap II) : Start with an unital associative algebra $A$ (over a field $F$) and ...
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Lie Bracket and Matrix Groups

The lie bracket appears in manifolds and matrix groups. For manifolds a tangent vector $X$ is $$X(p)=\sum_{i=1}^n a_i(p) \frac{\partial}{\partial x_i}$$ where there is the parametrization $\mathbf{x}:...
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Converse of Schur's lemma for Lie algebras. [duplicate]

The statement of Schur's lema for Lie algebras says that. Let $(\rho,\mathcal{V})$ is a complex irreducible finite-dimensional representation of a Lie algebra $\mathfrak{g}$. If $T$ conmmutes with $\...
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Deriving Lie Algebra for a Lie Group

I'm reading Helgason's DG, Lie Groups, and Symmetric Spaces and at one point he briefly mentions the Lobatchevski half-plane on page 136: The group $G$ of the mappings $T_{a,b}: x \to ax+b$ with $x ...
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Irreducibility of the derived representation

Let $G$ a linear Lie group with lie algebra $\mathfrak{g}$. If $(\pi, \mathcal{H})$ is a irreducible representation of $G$. Does the irreducibility of $\pi$ imply the irreducibility of derived ...
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82 views

Connected linear Lie groups $G$ generated by $\exp (\mathfrak{g})$.

Let $G$ be a connected linear Lie group with Lie algebra $\mathfrak{g}$. I understand that any open neighborhood of the identity of $G$ generates it, but, why does $\exp(\mathfrak{g})$ also generate ...
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Degree of universal cover of simple Lie group

I have seen a statement that if $\mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $\mathfrak{g}$. Equivalently, the simply connected group with Lie ...
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39 views

How to show that Symplectic group $Sp(n,\mathbb C)$ is not compact?

I wanted to prove above just using basic facts .I had only given that Sympletic group preserve following bilinear form : $B[x,y]=\sum_{i=1}^kx_iy_{n+i}-x_{n+i}y_i$ I had defination of compactness as ...
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Bilinear form on tensor product restricted to direct summands

Let $\mathfrak{g}$ be a complex semi simple Lie algebra. Then $\mathfrak{g}$ is equipped with a canonical $\mathfrak{g}$-invariant non-degenerate bilinear form $\beta$. Now this gives a $\mathfrak{g}$-...
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Lie group map whose differential is an isomorphism is a covering map

While trying to read the proof in Fulton and Harris of their “Second Principle,” I ran across something that I do not understand. They seem to claim that if $f: G\rightarrow H$ is a map of Lie groups ...
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Adjoint functors for Lie Algebras

Let's restrict to finite dimensional case. Functor $(-)^{\mathrm{ab}}: \mathrm{LieAlg} \to \mathrm{AbLieAlg}$ is left adjoint of the inclusion functor $i: \mathrm{AbLieAlg} \to \mathrm{LieAlg}$. ...
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Differential form ''equation''

I am having a bit of a trouble with the following. I'm working in the homogeneous Lie Group $\mathbb{R}\ltimes \mathbb{R}^3$ with an specific bracket an it give me de following system to integrate and ...
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1answer
38 views

Connection between a Lie algebra's root system and it's Lie bracket

I have for some time been trying to understand what the root systems of Lie algebras "mean". I understand that vaguely speaking, the Lie algebra is the derivative of the corresponding Lie group at the ...
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1answer
28 views

Determinant of adjoint representation

Let $G$ be a semisimple Lie group with Iwasawa decomposition $G=KAN$ and consider the determinant of the adjoint representation $\operatorname{Ad}$ of $AN$. I want to determine what the derived ...
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1answer
87 views

Existence of a semisimple elliptic subalgebra

Let $\mathfrak{g}$ be the Lie algebra of a compact Lie group $G$. Denote by $\mathbb{C} \mathfrak g = \mathbb{C} \otimes \mathfrak g$ the complexification of $\mathfrak g$. If necessary, I would not ...
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Definition of the Lie algebra and the Lie bracket for general vector fields

I've started to go deep into the theory of Lie groups to eventually understand their representation theory. I picked up a text online and right on the first chapter something started to bother me. ...
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Existence of a non essentially real ideal in a semisimple Lie algebra

Let $\mathfrak{g}$ be the Lie algebra of a semisimple compact Lie group $G$. Denote by $\mathbb{C} \mathfrak g = \mathbb{C} \otimes \mathfrak g$ the complexification of $\mathfrak g$. I am assuming ...
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Decomposing $k$th exterior powers $\Lambda^kV(\omega_1)$

Let $\Phi$ be a $G_2$ root system, $\omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V = V(\omega_1)$ of highest weight $\...
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1answer
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Unique faithful $7$-dimensional representation of semisimple Lie Algebra with $G_2$ root system

I am asked to show that if $\mathfrak{g}$ is a semisimple Lie Algebra with root system of type $G_2$, then it has a unique, $7$-dimensional faithful representation. To start, let $\omega_1, \omega_2$...
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If two rotation matrices commute, do their infinitesimal generators commute too?

Suppose that $e^A$ and $e^B$ are two rotations in $\mathrm{SO}(n)$. If $e^{A}e^{B} = e^{B}e^{A}$, can we conclude that $e^{A+B}=e^Ae^B$? More importantly, can we say that $AB=BA$? I'm particularly ...
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Highest Weights of Defining and Adjoint Representations of $\mathfrak{so}_5$

I am asked to describe the defining representation of $\mathfrak{sp}_4$ in terms of highest weights, and then I am asked to repeat this process for the defining and adjoint representations of $\...
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1answer
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Understanding proof of PBW 1

This is first of series of questions I have in understanding PBW. I am following this note. Ring $R$ is assumed to be commutative. Definition 1: Let $\mathfrak{g}$ be a Lie algebra over a ...
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Irreducible Dual Representation

For a semisimple Lie Algebra $\mathfrak{g}$ with Cartan Subalgebra $\mathfrak{t}$, let $V(\lambda)$ be the unique irreducible highest weight module with highest weight $\lambda$. I am asked to show ...
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Calculating the Formal character on the irreducible $(n+1)$ dimensional representation of $\mathfrak{sl}_2$

Let $V(n)$ be the unique, irreducible representation of $\mathfrak{sl}_2$ of $(n+1)$-dimensions. Let $\rho$ be the sum of all fundamental weights. I want to calculate the formal character $ch(V(n)) ...
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The Character/Weight of a Representation of an Algebra

This may well be something of a silly question, but if so, then all the more reason I get it straightened out. I have in the past been working with representations of both groups and algebras, and in ...
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$SU(2)$ and its representations

Does the Lie-Algebra of $SU(2)$ always have 3 generators? Because I'm reading about different representations of $SU(2)$ as $2\times2$, $3\times3$, $4\times4$ matrices etc. But I guess that even if we ...
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1answer
28 views

Subgraphs of Dynkin Diagrams

Am I right in thinking that if we have two semisimple Lie Algebras $\mathfrak{g} $ and $\mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras ...
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Which roots are fixed by simple reflections of the Weyl Group?

Let $\Phi$ be a root system of a semisimple Lie Algebra, and $W$ it's Weyl group. Let $\Delta = \{ \alpha_1, \dots, \alpha_l \}$ be a root basis, and let $w_i \in W$ be the simple reflection ...
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1answer
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Is this pseudo-Cartan decomposition of $SO(n)$ valid?

I'm a graduate student in a field of science where we frequently need to optimize a matrix in $U(n)$ or $SO(n)$ (henceforth $SO(n)$ for concreteness) to get an "optimal" orthonormal basis before doing ...
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Lie algebra generator that does not appear on rhs of the algebra

Is it possible that in a lie algebra one may have a generator that does not appear on any of the commutators? If the above line is too physicisty for you, maybe I can try to translate to a more ...
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1answer
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$\frak{g}^{\bot}$ $=Z(\frak{g})$ if and only if $\frak{g}$ is reductive

Let $\frak{g}$ be a finite dimensional Lie algebra over $k$, a field of characteristic $0$. Recall that $\frak{g}$ is called reductive if the center $Z(\frak{g})$ is equal to the radical, rad$(\frak{...
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The intersection of a maximal toral subalgebra with a simple ideal of a Lie algebra is a maximal toral subalgebra of the simple ideal.

I'm reading Humphreys' Introduction to Lie Algebras and Representation Theory and I have a question about Corollary 14.1, which reads: Humphreys Corollary 14.1. Let $L$ be a semisimple Lie algebra,...
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Prove that $[\frak{g}$ $,cen(I)] \neq cen(I)$

Let $\frak{g}$ be a finite dimensional Lie algebra over an algebraically closed field, $k$, of characteristic $0$. Suppose that $I \triangleleft \frak{g}$ is an ideal of co-dimension $1$, and that $\{...
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Is there a relationship between pre-Lie algebras and post-Lie algebra?

You can find a short survey in this paper https://arxiv.org/pdf/1712.09415.pdf on Post-Lie algebras. I am interested in them because both of them can be constructed on rooted trees. But I don't know ''...
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Classification of metric Lie Algebras of dimension 7?

I want to classify all metric two-step Lie algebras of dimension 7. Clearly, the dimension of the center of such Lie algebras is $\leq 5$. So, I have to consider 5 cases separately, where the ...
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The lattice generated by $\{w(\rho) - \rho\,\vert\,w\in W\}$

Consider an irreducible root system associated to a complex simple Lie algebra $\mathfrak{g}$. Let $\rho$ be the half sum of positive roots and let $W$ be the Weyl group. Then what is the lattice $L$ ...
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Derived series of Lie algebra in reverse

Consider the first element in the derived series of a Lie algebra $L$, defined by $L^{(1)}:=[L,L]$. For a given Lie algebra $\tilde{L}$, is there always a Lie algebra $L$ such that $L^{(1)}=\tilde{L}$...
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Reference for Jacobson's theorem about lie algebras

My book (Lie groups by Postnikov) has the following theorem, which it calls Jacobson's theorem: Let $A$ be a unital associative algebra over some field of zero characteristic, $X$ a subset of $A$ ...
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114 views

Lie algebra of a matrix group

Let $G$ be a Lie group of matrix. I can define two Lie algebras from there : $G'$: the set of matrices obtained by computing componentwise the derivative $\gamma'(0)$ of every paths $\gamma$ in $G$ ...
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1answer
56 views

Effect of a Lie group morphism associated to a Lie algebra morphism to wedge product

I currently struggeling with the last exercise on my assignment: Fix $\omega\in\bigwedge^3(\mathbb R^n)^*$. Let $G$ be a Lie group, $\rho\colon G\to GL(n,\mathbb R)$ a Lie group morphism such that $$ ...
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Modifying unitary matrix eigenvalues by right multiplication by orthogonal matrix

I have a matrix $U \in U(n)$ ($U^* U=Id$), with eigenvalues $\lambda_1, \dots \lambda_n \in S^1$. I would like to know if its always possible to find a matrix $O \in O(n)$ such that the eigenvalues $\...
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Irreducible highest weight representations as a graded algebra

Let $L$ be a semisimple Lie algebra and let $V(\lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $\lambda$. How can we view the sum \begin{align*} \oplus_{n\in\mathbb{N}...
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Linear map connecting two left-invariant one-forms valued in different Lie algebras

How to see that a left-invariant one-form on a Lie group valued in a different Lie algebra can be factorized through the canonical left-invariant Maurer-Cartan form of this Lie group followed by a ...
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Is the vertex algebra associated to a negative definite even lattice a vertex operator algebra?

Is the lattice vertex algebra associated to a negative definite even lattice a vertex operator algebra? In particular if $L=\mathbb{Z}\alpha$ with $<\alpha, \alpha>=-2$, does $V_L=\bigoplus_{n \...
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If $(e^{\alpha(t)})'=\alpha'(t)e^{\alpha(t)}$ then $\alpha(t), \alpha'(t)$ commute?

Let $\alpha(t)$ be a smooth path of real $n \times n$ matrices. (Formally $\alpha:(-\epsilon,\epsilon) \to M_n(\mathbb{R})$). If $(e^{\alpha(t)})'=\alpha'(t)e^{\alpha(t)}$ for every $t$ or $(e^{\...
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1answer
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One dimensional image of the adjoint action

Let $K$ be an algebraically closed field of characteristic zero, and let $\mathfrak{g}$ be a finite dimensional, nilpotent Lie algebra over $K$. My question is, can we find an element $x\in\mathfrak{...
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Normal subgroup in a matrix Lie group

Prop: If $G$ is a Matrix Lie group, then the connected component that contains the identity $I$ is a normal subgroup of $G$. I have problem in the proof of this. Suppose that $A$ and $B$ belong to ...
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Lie algebra generator relation $T^a T^b \propto T^c $ valid for any $a,b$?

Given a Lie Algebra (such as $su(n), so(n))$ can I always find a set of generators + identity $\{T^a\}\cup \{id\}$ such that there exists a $c$ for any given $a,b$ such that $T^a T^b = C(a,b) T^c $ ...
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Killing form for a different representation

In this Math Overflow question, the OP was asking if the Killing form defined for a different representation (than the adjoint) was related to the normal Killing form (defined w.r.t. the adjoint rep.):...