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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What is the image of an automorphism of Lie algebra?

Let $L$ be a simple Lie algebra over ${\rm GF}(2)$. If $α$ is an automorphism of $L$ then for any element of $L$ we must have $α[a,b]=[α(a),α(b)]$. Now I want to have a clear understanding of image of ...
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Applications and usefulness of universal enveloping algebra

I know the definition of the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, and I know the PBW theorem. My question is the following: Where does the concept of the universal ...
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21 views

What is a conjugate weight?

The authors here write that the longest element of the Weyl group is $$w_{\max} = - id$$ except for $E_6$, $A_r$ and $D_r$ with $r$ even. There they write that $w_{\max}$ acts on a weight $\lambda$ ...
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7 views

Does complexification make a self-conjugate representation non-self-conjugate?

I recently learned that a non-self-conjugate representation is not the same as a complex representation. Given a real representation $\pi$, with highest weight $\mu$ $$\pi : \mathfrak{g} \rightarrow ...
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1answer
16 views

Show that a one dimensional $\mathfrak {g}\!-\!\operatorname{module}$ is irreducible

I want to show that a one dimensional $\mathfrak {g}\!-\!\operatorname{module}$ is irreducible. My problem comes down to undestanding the basis of this object. Now assuming the basis works in the ...
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20 views

Does the Weyl group act on its Lie algebra?

I am trying to prove something about the action of a particular Lie algebra on a particular representation (it's the starred claim on page 7 of this paper for those interested). My friend showed me a ...
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2answers
48 views

$\mathfrak{sl}(2)$ is a simple Lie algebra.

I am trying to prove that $\mathfrak{sl}(2,\Bbb C)$ is simple. Since this takes the $[x,y]=xy-yx$ matrix commutator bracket, this is clearly non-abelian. So to prove it is simple, we need only show ...
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Why do “the Dynkin components of a weight play the role of eigenvalues with respect to the generators $H^i$ of the Cartan subalgebra”?

In the book "Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists" by Jürgen Fuchs,Christoph Schweigert the authors write "In the description of representations, the ...
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31 views

Is there some connection between the killing form and a killing vector field?

I'm studying a paper on globally symmetric spaces, and the methods involved use at some points Killing vector fields. Since globally symmetric spaces can be well-understood by methods of Lie algebras, ...
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1answer
26 views

General Element of U(4)

Relating back to previous question about how to write a general element of $U(2)$, I am now wondering about how to write a general element of $U(4)$. Define $\Gamma_{(i,j)}:=\sigma_i\otimes\sigma_j$ ...
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1answer
18 views

Is the fundamental weight basis (a.k.a Dynkin basis) an orthonormal basis?

The simple root $\alpha_i$ basis is not an orthonormal basis, as can be seen from the Cartan matrix, which encodes how much they aren't orthonormal. For simplicity, let's assume a simply-laced Lie ...
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Understanding $G_2$ inside Spin(7)? (EDIT: problem solved)

This is a rather embarrassing question, so please let me know of any duplication and I will happily remove it. I am seeking to understand the $\mathbb Q$-split form of the algebraic group $G_2$, and ...
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18 views

In what sense are complex representations of a real Lie algebra and complex representations of the complexified Lie algebra equivalent?

In this book I read Proposition A.1. The irreducible complex representations of a real Lie algebra $\mathfrak{g}$ are in one-to-one correspondence with the irreducible complex-linear ...
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1answer
49 views

Are the physics and math definitions of a complex representation equivalent?

I was astonished to read at Wikipedia that The term complex representation has slightly different meanings in mathematics and physics. In mathematics, a complex representation is a group ...
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25 views

Obtaining Cartan Matrices from Lie Algebras

I have been studying Lie Algebras from chapter 13 of Conformal Field Theory by Di Francesco et al., I understand that the entire structure of a Lie algebra is contained in its Cartan Matrix. What I do ...
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28 views

What does it really mean to complexify the $10$-dimensional representation of $ \mathfrak{so}(10)$?

A commonly used "trick" in $SO(10)$ Grand Unified Theories is to use a "complex" instead of a "real" $10$-dimensional representation for the Higgs fields. My problem is understanding what this ...
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2answers
32 views

Does the matrix logarithm always converge for exponential matrices?

We have defined functions on square matrices $X$: $$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$ and $$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$ The exponential converges for ...
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11 views

Complexification of a Lie algebra representation in terms of weights?

EDIT: I found in this book the sentence: The weight system of a real representation of $G$ is defined to be the weight system of its complexification I think if someone can explain what this ...
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0answers
37 views

Why is a weight automatically a complex weight?

EDIT: I think the partial answer to my question is that in order to talk about weights we always need a complex Lie algebra. If the Lie algebra is real, we use the complexification, This is necessary, ...
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36 views

Where to the degrees of freeedom go when a complex representation becomes a real representation of a subalgebra?

As an example consider the complex $16$-dimensional representation of $\mathfrak{so}(10)$. When $\mathfrak{so}(10)$ is reduced to the subalgebra $\mathfrak{so}(9)$, the complex $16$-dimensional ...
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7 views

Regular elements on Lie algebra and on Lie group level

I want to understand (better) the definition and meaning of regular elements of semisimple Lie groups resp. Lie algebras. A regular element $X$ of $\mathfrak g$ is one whose centralizer has smallest ...
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23 views

An incorrect proof that the Lie algebra matrix exponential is always injective. What's wrong?

Suppose we have two square complex matrices $X,Y $ in the lie algebra $\mathcal{G}$ of matrix Lie group $\mathbb G$ such that $e^X = e^Y$. Then $e^{tX}$ and $e^{tY}$ define the same one-parameter ...
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1answer
184 views

Lie bracket; confusing proof from lecture

I am having some difficulties understanding this proof. Let $G$ be a closed matrixsubgroup of the general linear group. We have a right translation $Y(g):=dR_g(e) Y(e)$ on the Lie algebra $Y \in ...
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1answer
21 views

S-subalgebra and R-subalgebra.

Given the characteristics of a Cartan subalgebra H of a semisimple Lie algebra; H is abelian, it stabilises the root spaces generated with respect to it. It appears to me that all subalgebras of a ...
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17 views

Affine Weyl group as coxeter group

How do you write the affine Weyl group corresponding to type $A_n$ as a Coxeter group ?The generators are $s_0,s_1,s_2,\cdots ,s_n$ where $s_0$ corresponds to the highest root. What are all the ...
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1answer
45 views

Lie algebra: Efficient way of finding commutation relations

Is there an efficient way of finding commutation relations for a Lie algebra? For $\mathfrak{su}(2)$ with the Pauli matrices multiplied by $-\frac i2$ we get only three non-trivial commutation ...
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4answers
110 views

Show that Lie bracket is in Lie algebra?

Let $so(3)$ be the Lie Algebra of $SO(3)$ and $R\in SO(3); \Omega_1,\Omega_2 \in so(3)$ and $\Omega_n = \frac{d}{dt}R_n(t)$ at the point $t=0$. So $\Omega_n$ is the tangent vector of the curve ...
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3 views

Outer automorphisms of semi-simple Lie algebras

It is known that outer automorphisms of semi-simple Lie algebras are automorphisms of their corresponding Dynkin diagrams. But would it be correct to say that for a semi-simple Lie algebra all outer ...
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1answer
17 views

Clarification on the non-abelian Lie algebra bracket (Dimension 2)

I have a dumb question: I have proven that there is one isomorphism class for all two-dimensional non-abelian Lie algebras, with basis $\{x,y\}$ and bracket $[x,y]=x$. and it was written in an ...
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142 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
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2answers
318 views

Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
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Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?

I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and ...
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0answers
40 views

Level sets on $SU(n)$

Given $G \in SU(n)$, what are the level sets of the function $F(V) = |tr(G^{\dagger}V)|^2$? Can they be written only in terms of abstract linear maps, not in terms of the components of $V$ and $G$ in ...
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1answer
53 views

Scalar product on Lie algebra of compact Lie group [duplicate]

I am studying Differential Geometry and I am facing with a lemma in which there is a step that I do not understand. In particular, let $G$ be a connected compact Lie group, is used "$\langle\ \cdot , ...
2
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1answer
34 views

Basis of Lie Algebra $\mathfrak{su}(3)$

With the matrices below, apparently $\{u_k = -\frac i2 \lambda_k| k=1,2,\cdots,8\}$ forms a basis of $\mathfrak{su}(3)$ How could that be true? $-\frac i2 \lambda_1$ shouldn't even be an element of ...
2
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1answer
29 views

Lie algebra vector subspace: Does $[n_1,[n_2,Y]]=[n_1,A]=B$

If $Y$ is a vector subspace of the Lie algebra $\mathfrak{g}$ and $n_1,n_2\in N_\mathfrak{g}(Y)$ does the following hold? $$[n_1,[n_2,Y]]=[n_1,A]=B$$ where $B\subseteq A\subseteq Y$
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1answer
28 views

Lie Algebra: Vector subspace is an ideal of its normalizer

Let $S$ be a vector subspace of the Lie algebra $\mathfrak{g}$. Is $S$ an ideal of the normalizer $N_\mathfrak{g}(S)$, I would say yes since: $\forall n\in N_\mathfrak{g}(S), [n,S]\subseteq ...
5
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1answer
47 views

Is a formal deformation of a Lie algebra an example of a formal group law?

I stumbled across the following definition of the formal deformation of a Lie algebra, and it looks like a group object in the category of formal schemes (not necessarily commutative or ...
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1answer
38 views

Subalgebras and Ideals of a Lie algebra

If $A$ is a vector subspace of $\mathfrak{g}$(Which is a Lie algebra), and $N=\{x\in\mathfrak{g}:[x,A]\subseteq A\}$ So if $N=A$, then $A$ is a subalgebra, and if $N=\mathfrak{g}$ then, $A$ is an ...
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1answer
589 views

Inscrutable proof in Humphrey's book on Lie algebras and representations

This is a question pertaining to Humphrey's Introduction to Lie Algebras and Representation Theory Is there an explanation of the lemma in §4.3-Cartan's Criterion? I understand the proof given there ...
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0answers
57 views

Solving a matrix differential equation

I am trying to solve: $\frac{d U_t}{dt} = Tr(G^{\dagger}U_t)G - Tr(U_t^{\dagger}G)U_t G^{\dagger} U_t$ Where $U_t \in SU(4)$ and $G \in SU(4)$ is given and constant. Is it possible to solve this ...
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1answer
27 views

Clarification of Fulton and Harris root lattices

Sorry to ask what is almost certainly a very trivial question, but in Fulton and Harris's first course in representation theory they write down a property of root lattices which I think must be sort ...
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2answers
27 views

$\mathrm{Z}(\mathfrak{gl}(2,\Bbb F))$ where the Lie bracket is $[X,Y]=XY-YX$

I want to find $\mathrm{Z}(\mathfrak{gl}(2,\Bbb F))$ where the Lie bracket is $[X,Y]=XY-YX$ So then this will depend on the field, but no harm in direct computation for arbitrary matrices: ...
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1answer
64 views

How to visualize $\operatorname{Lie}(\operatorname{GL}_n)=\mathfrak{gl}_n$ in positive characteristic?

I'd like an intuitive explanation as to why the Lie algebra of $\operatorname{GL}_n$ is $\mathfrak{gl}_n$ when working over fields of positive characteristic. Below I reproduce how I "see" this fact ...
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0answers
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Map for roots of a Lie group to roots of a special subalgebra?

For regular subalgebras $h$ of some group's Lie algebra $g$, $$ h \subset h $$ the root system of the subalgebra is a subset of the root system of the original's group algebra. Subalgebras whose ...
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1answer
101 views

Meaning of $[A,B]$ when $A$ and $B$ are self-adjoint

This is just a question about what constitutes standard notation. In a context where $A$ and $B$ are understood to be Hermitian matrices, what is usually meant by the bracket $[A,B]$ ? On the one ...
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1answer
34 views

Lie Algebra $[v,0]=0=[0,v], \quad\forall v\in \mathfrak{g}$.

I want to show that $[v,0]=0=[0,v], \quad\forall v\in \mathfrak{g}$. Is this done using the Jacobi identity? I am not sure how to do this, I just put $0$'s in the Jacobi identity, but it didn't give ...
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22 views

Generators for Lie algebra $\mathfrak{sp}$

I am doing a question on find a presentation for the Lie algebra $\mathfrak{sp}(n)$. My thought is first write down the generators and relations of $\mathfrak{sp}(n)$. Than write each generators in ...
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2answers
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Adjoint representation

I was just wondering why the adjoint representation of the Lie group $Ad$ and Lie algebra $ad$ are called representation. Maybe this word is derived from abstract algebra somehow, but I don't ...
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Local Lie derivative on $G$-space at the zero of a vector field

Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian). $L_X$ is a vector field on $M$, at a ...