For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

learn more… | top users | synonyms (1)

2
votes
2answers
238 views

Reference request: group theory

Currently I'm studying differential geometry and PDEs - so I often meet the use of groups. I also studied symmetries methods for solutions of differential equations but the connection between Lie ...
4
votes
1answer
317 views

How to obtain a Lie group from a Lie algebra

Please pardon me, if the question is too simple. How can we obtain a Lie group form a given Lie algebra (say in 3 dimension) in practise? Can someone illustrate it in 3 dimension? Is there a ...
2
votes
1answer
205 views

Algorithms to compute the orbit of the action of the Weyl group of a semisimple Lie algebra on a given weight?

Given a simisimple Lie algebra $\mathfrak{g}$ and a weight $\lambda$. Let $W$ be the Weyl group of $\mathfrak{g}$. Is there an algorithm (reference or software) to compute the set $W \cdot \lambda$ ...
0
votes
1answer
409 views

Highest root, highest weight and highest short root

Are highest root and highest short root the same? Are there some example to show that the highest root and the highest short are not the same? Are there some example to show that the highest root and ...
3
votes
0answers
252 views

Invariant inner product $\langle\,,\rangle$ on a Lie algebra

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We can use the Killing form to identify $\mathfrak{h}$ and $\mathfrak{h}^*$ ...
1
vote
1answer
149 views

Understanding: construction of standard cyclic L-module of highest weight $ \lambda$ by induced module construction

I am currently reading a text about the construction of a standard cyclic L-module of highest weight $ \lambda$ (L is a semisimple Lie algebra) and I am having trouble understanding the principle of ...
1
vote
1answer
150 views

Understanding: Common eigenvector of Borel subalgebra is a maximal vector

I didn't understand step b) of this proof and would be happy if someone could help me with this. Let dimV be finite. Let L be a semisimple Lie algebra, $\ L_\alpha $ a weight space. Let $\ \Delta $= ...
1
vote
1answer
119 views

Reference to a list of affine Cartan matrices

I can find a list of affine Dynkin diagrams in some books but cannot find a list of affine Cartan matrices. We can write down affine Cartan matrices using affine Dynkin diagrams. But are there a list ...
1
vote
1answer
169 views

Show that irreducible standard cyclic module is finite dimensional

I had problems understanding the following proof. Maybe someone could help me with this? Let {$\ x_i, y_i $} be the standard generators of the Lie Algebra L. Let $\ ...
0
votes
2answers
119 views

Lie Algebras: (ad $\ y)^4(z)=0 $, since root strings have length at most 4

Can somebody please explain this to me? (ad $\ y)^4(z)=0 $, since root strings have length at most 4. Note: y and z are root vectors belonging to two negative roots.
3
votes
1answer
199 views

fundamental representation of $\ sl(l+1,F)$

This problem concerns the topic representation theory of Lie Algebras. The main purpose of the exercise is to study the form of the fundamental dominant weights of a Lie Algebra. I would be very ...
1
vote
1answer
179 views

What is meant by “direct summand in a tensor product”?

I am currently working on the topic of Lie - Algebras and I have stumbled a few times over the expression "direct summand in a tensor product". The text says that $\ V(\lambda) $ as an ...
0
votes
1answer
178 views

What is meant by $\sum_{p + q = v + w} {\dim V_p * \dim W_q}$?

I am currently working on the topic of Lie - Algebras. What is meant by $\displaystyle\sum_{p + q = v + w} {\dim V_p * \dim W_q}$ ? $\ V_v $ and $\ W_w $ denote weight spaces I don't know how to ...
1
vote
1answer
181 views

Questions about Freudenthal's formula

I am reading the book Introduction to lie algebras and representation theory. I have some difficulty in understanding some parts of the book for Freudenthal's formula. Page 120, line 6, why ...
2
votes
2answers
168 views

Direct sum of an algebra and its opposite

I hate to do this, but I cannot seem to remember/find a particular result that I thought was true. Forgive me if I have some points wrong, since this is the point of my asking. I thought I remembered ...
5
votes
1answer
274 views

What is the basis for the Universal Enveloping Algebra of su(2)?

Given the standard basis for the Lie algebra $\mathfrak{su}(2)$ of SU(2), $\{i\sigma_1,i\sigma_2,i\sigma_3\}$ where $\sigma_1=\Biggl(\begin{array}{cc} 0&1\\ ...
1
vote
1answer
369 views

How to draw a weight diagram?

Given a weight, say $\omega=3\lambda_1+4\lambda_2$, where $\lambda_1, \lambda_2$ are fundamental weights (type A Lie algebra). How to draw the weight diagram of the irriducible representation with ...
3
votes
1answer
206 views

Universal Casimir element

On page 118 of J.E. Humphreys' book Introduction to Lie algebras and representation theory, paragraph 3 of section 22.1, what is the motivation of the definition of $c_{ad}$ in this way? Why we ...
7
votes
1answer
570 views

The mathematics behind Clebsch-Gordan Coefficients

In quantum physics we have to work a lot with Clebsch-Gordan coefficients and generalizations like the Wigner 3j,6j, and 9j symbols. In our coursework we are taught that the coefficients are coupling ...
1
vote
1answer
129 views

Question on the root systems

Let $\Phi$ be a root system of euclidean space $E$. Suppose that a subset $\Phi'\subset \Phi$ satisfies $\Phi'=-\Phi'$ and if $\alpha,\beta\in\Phi'$ and $\alpha+\beta\in \Phi$, then ...
5
votes
0answers
128 views

On the root systems

Let $\Phi$ be a root system of $E$. $\alpha,\beta\in \Phi$. Let $\lbrace \beta+i\alpha | i\in \mathbb{Z}\rbrace\cap \Phi$, $\alpha$-string through $\beta$, be $\beta-r\alpha,\ldots,\beta+q\alpha$, ...
3
votes
2answers
131 views

A certain subset of $\mathfrak{u}(n)$ is an embedded manifold?

I would have a hint on how to control if the following subset $C$ is an embedded submanifold of $\mathfrak{u}(n)$. $C$ is the set of the antihermitian $n\times n$ matrices $A$ with the property that ...
0
votes
1answer
115 views

some $\vee$ notation in lie algebras

Let $I$ be a set, $C=(c_{ij})$ be a generalized Cartan matrix, $r$ be the rank of $C$, $I'$ be a subset of $I$ such that $(c_{ij}), i, j \in I'$ is invertable. Let $\mathfrak{g}$ be a ...
2
votes
1answer
551 views

relations between root lattice and weight lattice

Let $Q$ and $P$ denote the $\mathbb{Z}$-span of the simple roots and fundamental weights respectively. What are the relations between $Q$ and $P$? Does $P$ contain $Q$? Thank you.
2
votes
1answer
521 views

Irreducible representations and simple lie algebras

Could you give me a hint how to prove the following? A representation $R$ of a complex Lie algebra $\mathfrak{g}$ is irreducible iff the image $R(\mathfrak{g})$ is a simple Lie algebra.
8
votes
1answer
277 views

Proof that Lie group with finite centre is compact if and only if its Killing form is negative definite

I am gathering material for an exposition on Lie theory and I am after proofs that a Lie group with finite centre is compact if and only if its Killing form is negative definite. I know of one, ...
1
vote
1answer
124 views

Connection 1-form on Lie group

If we regard $S^{2n-1} \to \mathbb{CP}^{n-1}$ as a principal $S^1$ bundle, how do I show that $$A=\frac{1}{2\pi}\sum_i(x_i dx_i-y_i dy_i),$$ where $(x_1,y_1,\dotsc,x_{2n},y_{2n})$ are coordinates on ...
2
votes
1answer
254 views

Casimir element of a universal enveloping algebra

Is the Casimir element of $U(sl_2)$ equal to $ef+fe+h^2/2$ or $(h+1)^2/4+fe$? Is $ef+fe+h^2/2$ equal to $(h+1)^2/4+fe$? How to compute the Casimir element? I think that $ef+fe+h^2/2 = ...
1
vote
1answer
297 views

Closed form for the exponential of a Lie algebra 3x3 matrix?

Matrices of the form: $$\begin{pmatrix} iz+iy&-iz+w&-iy-w\\ -iz-w&iz+ix&-ix+w\\ -iy+w&-ix-w&iy+ix\end{pmatrix}$$ where $x,y,z,w$ may be assumed to be real, form a Lie ...
1
vote
0answers
229 views

How to show that the structure constant of SU(3) is invariant?

So suppose $f_{ijk}$ is the antisymmetric structure constant of SU(3), and $D^8_{ij}(g)$ is the matrices of 8-dimensional adjoint representation of SU(3), then how to show that ...
0
votes
1answer
410 views

What is meant by adjoint of a linear transformation w.r.t a given inner product?

Consider a matrix Lie group equipped with a left &/or right invariant metric. The adjoint of linear transformation $A$ with respect to the inner product is denoted as $A^*$. Here what is ...
1
vote
1answer
242 views

Weyl Group, Lie algebra

How I can prove any element of order 2 in a Weyl group is the product of commuting root reflections. I need to show also that the only reflections in Weyl group are the root reflections.
2
votes
1answer
126 views

Relationship between Lie coalgebra and Lie bialgebra

I read the two Wikipedia articles and it sounds like there is a relationship between the two, but I can't quite grasp it. They don't seem to be the same thing, but I can't demonstrate it in part ...
11
votes
3answers
737 views

Lie algebra of a quotient of Lie groups

Suppose I have a Lie group $G$ and a closed normal subgroup $H$, both connected. Then I can form the quotient $G/H$, which is again a Lie group. On the other hand, the derivative of the embedding ...
3
votes
1answer
273 views

Why Lie algebras of type $B_2$ and $C_2$ are isomorphic?

both of Lie algebras of type $B_2$ and $C_2$ have dimension 10 and we can find two basis of them on page 3 in the book: Introduction to Lie algebras and representation theory . How could we show that ...
2
votes
0answers
95 views

How can I find a Chevalley basis of $B_2$?

How can I find a Chevalley basis of a type $B_2$ when the related lie algebra is defined as a linear Lie algebra of elements of the form $x= \begin{pmatrix} 0 & b_1 & b_2 \\ c_1 & m & ...
0
votes
1answer
177 views

How can I find a Chevalley basis of $B_2$ in the matrix realization of this group?

As is known, $B_2$ can be realized as linear Lie algebra of elements of the form $x= \begin{pmatrix} 0 & b_1 & b_2 \\ c_1 & m & n \\ c_2 & p & q \end{pmatrix}$, where ...
2
votes
2answers
241 views

What are the Weyl group of type $E_8$, $F_4$,$G_2$?

This problem is as titled. The textbook states that the order of the Weyl group of type $E_8$, $F_4$ are $2^{14}3^55^27$ and 1152 respectively, but I am wondering how are these groups like, namely, ...
2
votes
2answers
374 views

Is there any example of a Lie Algebra, who has nontrivial radical but contains no abelian ideal?

Is there any example of a Lie Algebra, who has nontrivial radical but contains no abelian ideal? Here, the radical of a Lie algebra means its maximal solvable ideal. This question occurs in the ...
5
votes
2answers
1k views

Classsifying 1- and 2- dimensional Algebras, up to Isomorphism

I am trying to find all 1- or 2- dimensional Lie Algebras "a" up to isomorphism. This is what I have so far: If a is 1-dimensional, then every vector (and therefore every tangent vector field) is ...
8
votes
1answer
359 views

Osp, USp, SU(,) and PSU

I would be glad if someone can give me some (hopefully easy to understand!) references for learning about these groups Osp, USp and PSU and their representations. I run into these mostly while ...
4
votes
1answer
111 views

How to compute the Gel'fand Models for a (quantum) Lie Algebra

Given a lie algebra $g$, how does one approach finding the Gel'fand models? For clarity, by this I mean $\bigoplus_{\lambda\in P^+}V(\lambda)$ where $P^+$ are the dominant weights, and ...
1
vote
1answer
180 views

Finding roots for a Lie algebra g, wrt toral subalgebra h

I'm trying to find the root space decomposition of a lie algebra wrt a toral subalgebra h. Both a matrix lie algebras. I'm confused about how do I find the linear forms $\lambda \in \mathfrak{h}^*$ ...
6
votes
3answers
2k views

How do you find the Lie algebra of a Lie group (in practice)?

Given a Lie group, how are you meant to find its Lie algebra? The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them? My group is the set ...
1
vote
0answers
116 views

Do the classical Lie algebras all satisfy $XM + MX^T = 0$?

I'm working on a homework assignment in which part of the question statement says that each of the classical Lie algebras can be described as the set of all matrices $X \in gl(n,\mathbb{C})$ ...
3
votes
1answer
816 views

Representation theory of $SO(n)$

This is probably not a very ethical question to ask but I need to have a fast introduction to a range of concepts about the representation theory of the $SO(n)$ and I would be happy to see some online ...
1
vote
2answers
99 views

$A \otimes_k A \to \bigwedge^2 A$ homs

Let $k$ be a unital commutative ring, and A be a $k$-module. Is there a homomorphism $f: A \otimes_k A \to \bigwedge^2 A$ such that $f(a \otimes a) \neq 0$ for some $a \in A$? I can take a hint :) ...
1
vote
2answers
110 views

An equality about characters of representations

This is an equality that I am gleaning out of some papers that I have been reading. I am not sure I am reading it right. Hopefully people will correct it. Let $U$ be a group element and $R$ be a ...
3
votes
3answers
335 views

Reasoning about Lie theory and the Exponential Map

I'm having a little difficulty wrapping my head around Lie theory (I'm a computer scientist, so perhaps that's to be expected). Specifically, considering the following definition from Wikipedia for ...
4
votes
1answer
264 views

does every adjoint orbit of a Lie group go through the Cartan subalgebra?

A naive question from a physicist, so forgive the lack of rigor. Consider a Lie group, acting on its Lie algebra by the adjoint action. Does every orbit go through the Cartan subalgebra? ...