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 derived algebra $[L,L]$ of a Lie algebra $L$

Definition of derived algebra of a Lie algebra $L$ is given by linear span of commutators $[x,y]$ for $x,y \in L$. but here why do we take linear span and why cant we just consider collection of all ...
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36 views

Nil and nilpotent restricted lie algebras

Let $k$ be a field of characteristic $p$, and let $L$ be a restricted Lie algebra over $k$. Thus $L$ is a lie algebra together with a map $(-)^{[p]}:L\to L$ satisfying the three axioms found here. ...
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88 views

semisimplicity of Lie algebra

Let $L$ be a lie algebra. Then if $L$ is semisimple, we have $L = L_1 \oplus \cdots\oplus L_n$ for some simple ideals $L_i$. But we can also consider the adjoint representation. In this ...
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481 views

video lectures on Lie algebra

Is there any video lecture on first course on Lie algebra available online? , by the first course I mean, The complete book of Introduction of Lie algebra and its representation theory by James ...
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45 views

gl(n,C) as a direct sum?

I've been trying to prove this: $gl_n(\mathbb{C})=Sl_n(\mathbb{C})\oplus \mathbb{C}I_n $ where $gl_n(\mathbb{C})$ is the General Linear Lie Algebra and $Sl_n(\mathbb{C})$ is the Special Linear ...
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137 views

Definition of Verma modules

I have a question regarding different (but equivalent!?) definitions of Verma modules of semisimple Lie algebras: Let F be a field and denote the following: $ \mathfrak{g}$ , a semisimple Lie ...
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2answers
39 views

Image of a nilpotent Lie algebra

Suppose $L_1$ and $L_2$ are two Lie algebras, and that $f: L_1\to L_2$ is a Lie algebra homomorphism. If $L_1$ is nilpotent, does it follow that $f(L_1)$ is nilpotent? Remark 1. The corresponding ...
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1answer
104 views

Equation On Root Systems (Humphreys Exercise 9.10)

I am stuck in the following problem from Humphreys. Let $\alpha, \beta$ be roots in a root system $\Phi$. Let the $\alpha$-string through $\beta$ be $\beta - r\alpha, \ldots, \beta + q\alpha$ and let ...
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122 views

Properties of Lie derivative

Let's have Lie derivative: $$ L_{V}\varphi = V^{\mu}\partial_{\nu}\varphi , \quad L_{V}A_{\mu} = V^{\nu}\partial_{\nu}A_{\mu} + (\partial_{\mu}V^{\nu})A_{\nu}. $$ How to show that for scalar and ...
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32 views

Finding an orthonormal basis for a gl(3) module

I'm trying to find an orthonormal basis for gl(3)-module V(ε1-ε3), where ε1-ε3 is the weight (1,0,-1) of the highest-weight vector. Using Gelfand-Tsetlin (/Zetlin/Zeitlin) patterns, I'm at the point ...
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71 views

What is the smallest Lie subalgebra of $ {{\frak{gl}}_{n}}(\mathbb{R}) $ whose center is the set of $ (n \times n) $-scalar matrices?

We know that the center of the Lie algebra $ {{\frak{gl}}_{n}}(\mathbb{R}) $ of all $ (n \times n) $-matrices is the Lie subalgebra of all $ (n \times n) $-scalar matrices. The Lie algebra $ ...
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108 views

When is the connected component of the identity of a matrix Lie group the union of the one parameter subgroups?

When is it the case that all elements in the connected component of the identity of a matrix Lie group can be written as a single exponential of some element of the corresponding Lie algebra?
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How can I show that $\mathfrak{sl}_n(\mathbb{C})$ is a simple Lie algebra?

The question is in the title: how can I show $\mathfrak{sl}_n(\mathbb{C})$ is simple? In every book I scoured, they say $\mathfrak{sl}_n(\mathbb{C})$ is simple but they do not provide a proof! Is ...
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194 views

Jordan–Chevalley decomposition

I'm trying to understand the proof of it in Humphreys(Humphreys 1972, Prop. 4.2, p. 17). And I've not got over which field we are working. The characteristic polynomial may not have roots in the ...
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1answer
79 views

Problem 4.4 - Lie Algebras - Humphreys

I read the exercise 4.4 in the book Introduction to Lie algebras and representation theory of J. Humphreys, and I do not quite understand the sentence : We start with $L\leq\mathfrak{gl}(p,F)$ as ...
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46 views

Lie algebra homomorphism

I'm sure I'm missing something really obvious here. This seems too stupid. On page 47 of Erdmann & Wildon's Introduction to Lie Algebras, we have the following set up. Let $L$ be a Lie subalgebra ...
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378 views

Does every Lie algebra come from commutator of some associative product operation?

Suppose $\mathfrak{g}$ is an Lie algebra. Is it possible to define an associative product operation $\star$ on $\mathfrak{g}$ such that $[A,B]=A\star B - B \star A$ ? If it is not possible to do so ...
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2answers
186 views

Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV

Let V be a real n-dimensional vector space. Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV. Note that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is a real vector space and is ...
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121 views

Faithfulness of adjoint representation of Lie algberas

Are there any simple or useful conditions (necessary & sufficient) under which the adjoint representation lie algebra is faithful ? One sufficient condition is semisimplicity, but perhaps this is ...
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89 views

Perturbation in characteristic p, or Why, really, does Lie's theorem fail?

While recalling some basics of Lie theory, I found a funny proof of the main lemma in Lie's theorem on triangularity of representations of solvable Lie algebras. It turns out that this proof has a ...
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Lie algebra associated to a linear form

Let $F$ be a finite dimensional vector space over a field $k$, if $f : F \to k$ is any linear form, I can define on $F$ a Lie algebra bracket by the following rule $$ [x,y]=f(x)y-f(y)x, $$ or in terms ...
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436 views

Dimension of the Lie algebra of $SO(n,k)$

Let $SO(n,k)$ denote the special generalized orthogonal group. Of course, $SO(n,k)$ is a Lie group. I know that the Lie algebra $so(n,k)$ of $SO(n,k)$ coincides with the Lie algebra of $O(n,k)$. I'd ...
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69 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
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161 views

Existence of an irreducible $L$-submodule

Suppose $L$ is a finite dimensional Lie algebra. Let $V$ be an $L$-module (i.e. $V$ is a vector space which $L$ acts upon). We are assuming that $V$ has a finite dimension. My question is the ...
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1answer
73 views

O(p,q; C) isomorphic to the usual orthogonal group O(p + q; C) for complex field

I'm trying to make sense of this statement that appears on wiki: "The group O(p,q) is defined for vector spaces over the reals. For complex spaces, all groups O(p,q; C) are isomorphic to the usual ...
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1answer
136 views

How to prove a property of Lie derivatives

I know that there are five properties for Lie derivative. But one of them I don't know how to prove. It is $ L_x[\omega(Y)]=(L_x \omega)(Y)+\omega(L_x Y)$ Note : Here $\omega$ is a covariant vector ...
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161 views

Derived algebra of a lie algebra contained in an ideal

Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{R}$ or $\mathbb{C}$. Assume $\mathfrak{i}$ is an ideal with $\mathfrak{g/i}$ abelian. Then the derived algebra $[\mathfrak{g},\mathfrak{g}]\subseteq ...
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1answer
107 views

The Poincare group: rank and casimirs.

It is often stated without proof in the particle physics literature that the Poincare group has rank $2$ and that as a result, the corresponding Lie algebra has exactly $2$ casimirs. How can one show ...
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182 views

Schur's lemma and Invariant subspaces of direct sums of irreducible representations

There is a corollary to Schur's lemma which says that : If $V$ is a finite dimensional irreducible complex representation of a group G or Lie algebra and $\phi :V \rightarrow V$ is an intertwining ...
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260 views

When is the Cayley transform of a matrix $J$-orthogonal?

The (real) general linear group is defined $GL(n)=\{A \in \mathbb{R}^{n\times n} \mid \operatorname{det}(A) \neq 0\}$. It is a matrix Lie group. Let $J$ be a constant $n$-by-$n$ real matrix. The ...
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249 views

Reference request for studying Lie group & Lie algebra representations

I am learning representation theory of Lie groups & Lie algebras from the book by Brian Hall. Unfortunately, this does not discuss infinite dimensional representations. Which books should I study ...
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152 views

Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
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59 views

Casimir Invariants within the universal enveloping algebra

I've been asked to determine the eigenvalue of the Casimir invariant $I_2$ on any irreducible module with highest weight $\lambda = (\lambda_1, \lambda_2, ..., \lambda_n)$, where; $$I_m = ...
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137 views

Sources for learning Lie groups and symplectic geometry for Quantum optics

I am asking this question on behalf of my junior who has recently joined in the graduate programme. As suggested by my boss, the student wants to work on quantum optics from a symplectic geometric ...
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1answer
183 views

exponential and Lie bracket

I've been reading that one can compute a form of the Baker-Campbell formula with direct computations through the power series of $\exp(X)$, $\exp(Y)$ and $\log(1+W)$, where $X$, $Y$ are non commuting ...
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1answer
102 views

How to find the representation of Lie algebra

I read a book about the Lie algebra, but I really don't understand the calculation of $ad(X)$. For example, we have a Lie algebra of bases: $$e_1=\left[\begin{array}{cc}1 & 0\\0 & ...
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126 views

How can these be the weights of the adjoint representation?

This is perhaps a stupid question. We consider $G =\text{SU}(3)$ and $\pi : G \to \textrm{GL}(\mathfrak{g})$ the adjoint representation that sends $g \in G$ to $Ad_g$ that acts on the Lie algebra ...
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66 views

Dynkin diagram construction

My question is how to construct the Dynkin diagrams of a semi-simple Lie group $G$, which is the product of simple Lie groups. Is it the combinaison of Dynkin diagrams of these simple Lie groups? For ...
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55 views

How to construct Dynkin diagrams for semisimple Lie algebras?

My question is: How can I construct the Dynkin diagrams of a semisimple Lie algebra $L$ which is the direct sum of simple Lie algebras, such as for example $\text ...
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1answer
253 views

Lie Algebras: How to compute the Killing Form on $\mathfrak{sl}_n(\mathbb{C})$ and Jordan Decomposition Theorem question.

I'm reading the Fulton and Harris Representation Theory book, trying to learn about Lie Algebras. On pg. 213, they compute the killing form on $\mathfrak{h}^*$ for $\mathfrak{sl}_n(\mathbb{C})$. I ...
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1answer
685 views

Action of a lie group on its lie algebra via the adjoint representation

I am a physics undergrad. The adjoint action of a group on itself is $\operatorname{Ad}: G \times G \to G$ is defined to be $\operatorname{Ad}:(g,h) \to g^{-1}hg$. The adjoint representation of the ...
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100 views

Kernel of the Lie bracket $[,]\colon\wedge^2\mathfrak g\to\mathfrak g$

I believe the following is probably well-known, but so far I couldn't find the answer by myself: Let $\mathfrak g$ be a real (finite-dimensional) Lie algebra, and $\wedge^2\mathfrak g$ its second ...
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216 views

Visualizing Lie groups.

I like to visualize lie groups as flows on some manifold. For example: $SO(2)$ can be visualized as rotations of $S^1$ and it's lie algebra as constant vector fields on $S^1$. Or $SO(1,1)$ can be ...
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32 views

Example showing that the product of ideals must be the span of the commutators

I'm trying to find an example showing why, in a Lie algebra, we can't just define the product of two ideals $I$ and $J$ to be the elements of the form $[x,y]$ where $x \in I, \; y \in J$. I imagine ...
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96 views

Easy proof that $\exp{Xt} = I \Rightarrow X = 0$

Let $X\in \mathbb{C}^{n\times n}$ and $I$ is identity matrix , than if: $$ \forall t\in \mathbb{R}\quad e^{Xt} = I $$ than $$ X = 0. $$ I'm looking for short and slick proof of this ...
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91 views

Concrete example of a particular 3-dimensional Lie algebra

I'm reading over the classification of 3-dimensional complex Lie algebras, and have come to the classification of a particular Lie algebra spanned by $\{x,y,z\}$ satisfying the relations $$[x,y] = y, ...
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Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
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65 views

Exterior Derivative Problem

Suppose $\theta$ is a differential $1$-form defined on a manifold and with values in the Lie algebra of a Lie group $G$. On $M\times G$ define the $1$-form $ad(g)\theta$ where $\theta$ is extended ...
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47 views

isomorphism of algebras and direct sum

Let $A$ and $B$ be (associative or Lie) algebras over a field $F$, and let $I \subseteq A$ be an ideal. Suppose we have $A/I \cong B$. Do we then have $A \cong B \oplus I$? The answer may be ...
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91 views

What does this dynamic system represent for?

I know systems like $$\frac{dx}{dt}=Sx$$ where $S$ is a symmetric matrix admit a solution that dialates along eigendirection of $S$. And systems like $$\frac{dx}{dt}=Ax$$ where $A$ is a ...