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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Suppose Del is root system. Then at least one simple component of Del is not isomorphic to A1 if and only if there is an embedding A2 to Del.

Suppose Del is root system. Then at least one simple component of Del is not isomorphic to A1 if and only if there is an embedding A2 to Del. where A1 and A2 are simple root system. The idea is There ...
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25 views

Root System in Lie algebra

What do you mean by "Simple Root System" in Lie algebra? What I undersatand what Simple root system is that if we can write each root "beta" say as beta equals to summation of integral coefficient ...
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2answers
31 views

Cartan subalgebra of semisimple Lie algebra

My question is: How can I construct the Cartan subalgebra of a semisimple Lie algebra L which is the direct sum of simple Lie algebras, such as for example su(2)⊕su(2)⊕su(2)?
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159 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
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1answer
30 views

Lie ideals of $gl_n(K)$

I am looking for some reference where I can find a detailed study of the Lie ideals of the general linear Lie algebra $gl_n(K)$ with the bracket $[A,B]=AB-BA$, where $K$ is a field (if there are ...
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16 views

Given basis for a Lie algebra, what is one for its Universal Central Extension

Given that I have an infinite basis for a Lie algebra $L$, and the information that $M$ is its Universal Central Extension, is $M$ unique? If so, what is the basis of $M$ in terms of that of $L$?
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Free Lie algebras and basis for a subpcae of a special degree

Let X^* be the the set of all words on basis elements of Lie algebra L and F is the vector space spanned by X^*. I do not know how can I define the basis elements and also the number of basis ...
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1answer
35 views

How can I describe Lie bracket for formal product of elements of Lie algebras

Let L be a Lie algebra with basis $B=\{x_1,...,x_{10}\}$, Is there any property to describe the following lie bracket: for example how I can decompose $[x_1 x_2 x_3 , x_5]=$? Here $x_1 x_2 x_3$ is ...
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17 views

A question on Cartan involution

Is there a real(or complex) Lie algebra $L$ for which the set of all involutions is an infinite commutative set but the center of $L$ is finite dimension space?(So the set of all Cartan involution ...
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1answer
57 views

explicit matrix example of irreducible representation of s0(3)

Can someone give me a concrete or an explicit example of an irreducible representation of the Lie algebra so$(3)$? I know they are given by the Wigner D matrices but I want an explicit example of such ...
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21 views

Dimension of a weight space which is of weight $0$.

Let $V$ be a module of a Lie algebra $\mathfrak{g}$ and $V_{0}$ be the weight space of $V$ of weight $0$. $$ V_0 = \{ v\in V: h.v = 0, h \in \mathfrak{h} \}, $$ $\mathfrak{h}$ is a Cartan subalgebra ...
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71 views

Why is $\mathfrak{sl}(n)$ the algebra of traceless matrices?

I'm studying Lie algebras as purely algebraic objects, without much of a background in the differential geometry surrounding Lie groups. The definition of $\mathfrak{sl}(n)$ has been given to me as ...
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21 views

How to compute $\lambda(h_i)$?

Let $\lambda$ be a weight and $h_i = h_{\alpha_i} \in \mathfrak{h}$, $\alpha_i$ is a simple root. $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra $\mathfrak{g}$. How to compute $\lambda(h_i)$? ...
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1answer
71 views

What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
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1answer
53 views

Identities involving adjoint action

I'm looking for list of identities involving adjoint action $\mathrm{ad}_A X = [A,X] = AX - XA$. For example, it can be easily shown that: \begin{equation} e^{\mathrm{ad}_A} X = e^A X e^{-A} ...
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2answers
55 views

Prerequisites to “Applications of Lie Groups to Differential Equations”

I'm currently a 4th year student at a university. I've become close with a professor and we talked about the topic of lie groups in differential equations. He then offered to do a reading course with ...
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2answers
55 views

The Weyl group of A_3

Could someone please list all elements of the Weyl group of the root system $A_3$ in terms of simple reflections. In this case the Weyl group is $S_4$. Its strange that GAP failed to list all elements ...
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1answer
36 views

Question on unitary representation of non-compact simple Lie groups

The following is an exercise appearing page 148 in Knapp's book, representation theory of semisimple groups. Let $G$ be a connected linear non-compact Lie group with simple Lie algebra $\mathfrak g$. ...
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34 views

Lie groups and Lie algebras via exponential

I'm having a hard time computing the (connected) Lie groups that have a given Lie algebra. For instance, if we know that the Lie algebra of a certain Lie group $G$ consists of matrices of the form ...
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0answers
16 views

Proof for a corollary from PBW theorem

I need to know how we can prove the following corollary : If $x_1, \ldots, x_n$ is a vector space basis for Lie algebra $L$ then a vector space basis for $U(L)$, $U(L)$ is universal enveloping ...
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25 views

$\mathfrak so(V,B)$ as subalgebra and trace if subsets of it.

I'm studying lie algebras, and got stuck on this one: Let $B$ be a bilinear form on a finite-dimensional vector space $V$ over $\mathbb F$. I've seen many books that say that $\mathfrak so(V,B)$ ...
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72 views

Invariant tensors in adjoint representation

Suppose we have a simple Lie group $G$ with algebra $\mathfrak{g}=\{X_a\}$, where the generators $X_a$ are in some matrix representation. Is it true that the only invariant rank $n$ tensor in the ...
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20 views

Do the Generalized Gell-Mann Matrices form a complete set?

Please bear with me, I'm studying Lie algebras as they are related to quantum mechanics, and most of my group theory knowledge is self-taught. I'm not sure how to prove this seemingly basic result. ...
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23 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
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67 views

Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
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22 views

bases in free associative algebras

I need an example and more details for the below notion: If X={x_1,x_2,...} be a set and by X^* we denote the set of all words named w of elements of X. Let F be a field and F be the vector space ...
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0answers
19 views

Trace functionals as invariant elements of $R[\mathfrak{g}]$ under $G$

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ and let $G$ be its inner automorphism group. Then $G$ acts on $R[\mathfrak{g}]\cong S(\mathfrak{g}^*)$ via $(\sigma\cdot f)(x) = ...
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123 views

Is it true that the commutators of the gamma matrices form a representation of the Lie algebra of the Lorentz group?

Wikipedia claims (http://en.wikipedia.org/wiki/Gamma_matrices): The elements $\sigma^{\mu \nu} = \gamma^\mu \gamma^\nu - \gamma^\nu \gamma^\mu$ form a representation of the Lie algebra of the ...
2
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1answer
33 views

Two actions of $U(\mathfrak{h})$ on $U(\mathfrak{g})$ where $\mathfrak{h}\hookrightarrow\mathfrak{g}$

Let $\mathfrak{h}$ be a Lie subalgebra of $\mathfrak{g}$, then by PBW theorem we know $U(\mathfrak{h})\hookrightarrow U(\mathfrak{g})$. Let $\{x_i, y_i\}$ be an ordered basis of $\mathfrak{g}$ where ...
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1answer
32 views

Adjoint Lie algebra homomorphism

I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
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21 views

Question on inner product on space of representations of compact Lie groups

Let $K$ be a compact connected Lie group, wiewed as subgroup of unipotent matrices. Let $G=\mathfrak{k}^\mathbb C$ be the complexification with Lie algebra $\mathfrak{g}=\mathfrak{k}\oplus ...
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definition of universal algebra and universal enveloping algebra

The basis of a universal algebra is a function b that takes some algebra elements as values b(i) and satisfies either one of the two equivalent conditions named Outer condition and Inner condition ...
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Quadratic Casimir of fundamental irreps of simply-laced Lie algebras [migrated]

I have the following question, motivated by the expression for the character of level 1 highest weight integrable representations of simply-laced affine algebras (in terms of the string function). It ...
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28 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
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2answers
43 views

Classify Lie algebra with 1 dimension derived algebra

Problem 3.10 from Erdmann, Wildon ask: Find, up to isomorphism, all Lie algebras with 1-dimensional derived algebra. (also, this book assume finite dimension Lie algebra only; I'm not sure whether ...
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29 views

Angles between adjacent roots in a reduced root system.

Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer). ...
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1answer
31 views

Is it true if $[LL] = L$ then $L$ is a semisimple Lie algebra?

Let $L$ be a finite dimensional Lie algebra over $\mathbb{C}$. It is classical theorem that if $L$ is semisimple, then $[LL] = L$. Is it true if $[LL] = L$ then $L$ is a semisimple Lie algebra? I've ...
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Finding a basis and weight space for $L = so_6(\mathbb{C})= \{x \in End(\mathbb{C}^6)|^txS + Sx = 0 \}$

The question: Let $S = \left(\begin{array}{cc} 0 & I_3 \\ I_3 & 0 \end{array}\right)$ and let $$L = so_6(\mathbb{C})= \{x \in End(\mathbb{C}^6)|^txS + Sx = 0 \}$$ 1) Find a basis for $L$ ...
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1answer
32 views

Confusion regarding PBW theorem

I was reading up Humphrey's Introduction to Lie Algebras and Representation Theory and have a confusion regarding a consequence of PBW. First some notations: Let $L$ be a Lie algebra over ...
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Let‎$‎ L =L_0 ‎\dotplus‎ L_1‎$‎ be the Fitting decomposition of‎$ ‎L‎‎$‎ relative to‎$‎ adx‎$‎.Why then ‎$‎L_1=‎\bigcap‎_{i=1}^{‎\infty} L(adx)^i‎$‎?

Let‎$‎ L =L_0 ‎\dotplus‎ L_1‎$‎ be the Fitting decomposition of‎$ ‎L‎‎$‎ relative to‎$‎\mathrm{ad}(x‎)$‎.Why then ‎$‎L_1=‎\bigcap‎_{i=1}^{‎\infty} L(\mathrm{ad}(x‎))^i‎$‎?
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Induced Lie algebra homomorphism from Lie group homomorphism: left translation

A general result of Lie Theory is that every Lie group homomorphism $\Phi: G\rightarrow H$ induces a Lie algebra homomorphism $\phi: \frak{g} \rightarrow \frak{h}$. Which Lie algebra homomorphism ...
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1answer
146 views

Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
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automorphism group of Lie algebras

According to the difinition of automorphism group of lie algebra we must have the following condition: f [a,b]=[f(a),f(b)] for f in Aut (L) and L is a Lie algebra Now I have computed the automorphism ...
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1answer
39 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
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53 views

Regular elements of a module is open and dense

Let $\mathfrak{g}$ be an algebraic Lie algebra and $V$ be a $\mathfrak{g}$-module, then for each $v\in V$, define $\mathfrak{g}^v = \{x\in\mathfrak{g}:xv = 0\}$. Let $V_{reg}$ be the set of all $v$ ...
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1answer
91 views

Itzykson-Zuber integral over orthogonal groups

I would like to know is there a closed form expression for the following Itzykson-Zuber integral for the orthogonal case. $I=\int_{\mathcal{O}(p)} ...
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23 views

What are the root systems for the n-dimensional torus?

My question may seem silly at first, but currently I am not able to work out the question of finding all roots for the n-dimensional torus. At first, it seemed obvious to me that there are no roots at ...
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20 views

invariant polynomial on a lie algebra $\mathfrak{g}$

This question (maybe an easy one) arose when I was reading Humphrey's book "an introduction to Lie algebra and its representations". Suppose $\mathfrak{g}$ is a complex semisimple lie algebra, $V$ ...
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13 views

what is the image of an automorphism of lie algebra

Let suppose L be a simple lie algebra over GF(2) , if α be 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 α. Since ...
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prove if ‎$\frac{L}‎\varphi‎(L) S\bigoplus‎ R$ then ‎$ L\in\mathfrak{X} $

‎‎‎‎‎prove if ‎$\frac{L}\varphi‎(L) ‎‎‎S\bigoplus‎ R$ ‎where‎ ‎$ ‎\varphi‎(L) $ ‎is ‎the ‎frattini ‎ideal ‎of ‎$L$, $S$ ‎‎is a‎ $‎3$-dimensional ‎simple ‎ideal ‎of‎ ‎$\frac{L}\varphi‎(L)‎$ isomorphic ...