Tagged Questions

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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52 views

How to visualise the Killing form of a Lie algebra

Given a Lie algebra $\mathfrak{g}$, we can define its Killing form $$K(x,y) = \mathrm{Tr}(ad_x\circ ad_y)$$for $x, y\in \mathfrak g$. Whilst I understand that the Cartan decomposition $$\mathfrak g ...
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1answer
28 views

Killing form and Roots

I know that the roots of a Lie Algebra are functionals such that if $\alpha$ is a root and $h \in \mathfrak h$ is an element of the Cartan subalgebra, then $\alpha(h)$ is an eigenvalue. I'm looking ...
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0answers
55 views

Root space decomposition

Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the ...
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0answers
30 views

Toral sub algebra

It seems to me, I could be wrong, that the toral sub algebra goes against the following rules: For a semisimple Lie algebra: If the killing form is nondegenerate the Lie algebra is semi simple-> the ...
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1answer
33 views

Root space question

Do the roots of a root space decomposition have a kernel? Since it is the duel space to the cartan subalgebra ,evaluation of the roots on a non-equal index cartan basis element should be zero. Thanks ...
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33 views

Root Space Decomp

I am reading Humphrey's intro to Lie algebra, self-teaching, and have a few questions regarding root space decomp. 1) If I understand this correctly, the toral sub algebra of L represents all ...
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0answers
62 views

Prove the Weyl's complete reducibility Theorem on finite-dimensional $\mathfrak{g}-modules$ by Kostant's $\mathfrak{n}$-cohomology result

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)). But I have no idea about its proof. Any ...
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1answer
27 views

Let ‎$‎L‎$‎ be a Lie algebra. why if ‎$‎L‎$ ‎be‎ supersolvable then ‎$‎L'=[L,L]‎$ ‎ is nilpotent.‎

Let ‎$‎L‎$‎ be a Lie algebra. why if ‎$‎L‎$ ‎be‎ supersolvable then ‎$‎L'=[L,L]‎$ ‎(derived algebra) is nilpotent.‎
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58 views

Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...
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1answer
41 views

An extension of an algebraic question from my test

Let $A$ and $B$ be two real $n\times n$ matrices s.t. $AB=BA$. We now that $\det(A^2+B^2) \geq 0$. Is the similar question true for $n$ matrices which commute with each other? If not, how do I ...
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16 views

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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1answer
51 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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2answers
189 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
36 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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0answers
17 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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0answers
7 views

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
45 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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0answers
22 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 ...
2
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1answer
100 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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0answers
26 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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2answers
117 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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0answers
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
93 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
56 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
80 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 ...
2
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2answers
73 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 ...
3
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1answer
59 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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0answers
21 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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0answers
28 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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0answers
90 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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33 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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0answers
37 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) ...
3
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1answer
102 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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25 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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2answers
152 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
38 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
45 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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0answers
23 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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16 views

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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0answers
35 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 ...
2
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2answers
48 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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0answers
32 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
37 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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0answers
32 views

Finding a basis and weight space for $L = \mathfrak{so}_6(\mathbb{C})= \{x \in \operatorname{End}(\mathbb{C}^6)\ |\ ^{\mathrm t}xS + Sx = 0 \}$

The question: Let $S = \left(\begin{array}{cc} 0 & I_3 \\ I_3 & 0 \end{array}\right)$ and let $$L = \mathfrak{so}_6(\mathbb{C})= \{x \in \operatorname{End}(\mathbb{C}^6)\ |\ ^{\mathrm t}xS + ...
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1answer
37 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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1answer
63 views

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 ...
8
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1answer
216 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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1answer
46 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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0answers
55 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 $v$ ...
3
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5answers
179 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)} ...