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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27
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4answers
1k views

“Cayley's theorem” for Lie algebras?

Groups can be defined abstractly as sets with a binary operation satisfying certain identities, or concretely as a collection of permutations of a set. Cayley's theorem ensures that these two ...
17
votes
4answers
2k views

Jacobi identity - intuitive explanation

I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as ...
12
votes
2answers
359 views

Why are knot invariants best organized as polynomials?

Does anyone have a good explanation for why Knot invariants tend to be well organized as polynomials? What exactly is going on and why don't we often see polynomial invariants for classifying other ...
12
votes
1answer
219 views

How to prove these Lie algebra relations

This is a bit of a basic computational question concerning Lie algebras, but I'm getting kind of bamboozled so I thought I'd post it. I'm confused about how to perform some computations in Serre's ...
11
votes
3answers
789 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 ...
11
votes
2answers
739 views

References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the ...
11
votes
1answer
636 views

Exercise 6.5 in Humphrey's Book on Lie Algebras

I am trying to solve Exercise 6.5 part 4 in James Humphreys' Introduction to Lie Algebras and Representation Theory. I added the (homework) tag because my question is about an exercise, but this is ...
10
votes
3answers
246 views

Translations in two dimensions - Group theory

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
10
votes
5answers
225 views

Does non-commuting $\mathfrak{g}$ imply non-abelian $G$?

Question 8.1 in Kristopher Tapp's introductory text on matrix groups asks to show that $SO(n)$ is non-abelian ($n>2$) by finding two elements of $so(n)$ that do not commute. Why is this method ...
10
votes
2answers
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 ...
10
votes
1answer
126 views

Is it really unknown that every endomorphism of the Weyl algebra $A_1$ is an isomorphism?

Here $A_1 := K\{x\cdot-, \frac{d}{dx}\} \subset \operatorname{End}_K(K[x])$ for some characteristic-zero field $K$. I found this claim in Coutinho's "A Primer of Algebraic D-Modules." If this is ...
10
votes
1answer
77 views

Cauchy gave 1st example of a Lie algebra in 1847 & exterior product in 1853‽

I read in PDF pg. 5 of this that Cauchy gave the first example of a Lie algebra in 1847: It also claims that he invented the exterior product in 1853. Does anyone have references for this?
9
votes
3answers
272 views

What is the main use of Lie brackets in the Lie algebra of a Lie group?

I am beginner in Lie group theory, and I can't find the answer a question I am asking myself : I know that the Lie algebra $\mathfrak g$ of a Lie group $G$ is more or less the tangent vector of $G$ at ...
9
votes
2answers
924 views

On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
9
votes
2answers
111 views

Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $

I've been trying to find a Lie algebra isomorphism $$\mathfrak o(4,\mathbb R)\cong\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $$ but haven't managed so far. I have written down the ...
9
votes
2answers
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 ...
9
votes
1answer
119 views

Category of Lie group representations equivalent to the category of representations of their Lie algebra

Let $G$ be a lie group and $\mathfrak{g}$ its lie algebra. Consider the category $Rep(G)$ of finite dimensional representations of $G$ and the category $Rep(\mathfrak{g})$ of finite dimensional ...
9
votes
1answer
269 views

The center of a simply connected semisimple Lie group

I am learning about Lie groups, and I have the following basic question: Every Lie group $G$ has a (unique) universal covering group $ \bar G $ that is simply connected, and such that the covering ...
9
votes
2answers
310 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
9
votes
0answers
112 views

$\Delta \subset \Phi$ is a base in a root system imples $\Delta^\vee \subset \Phi^\vee$ is a base in a root system [duplicate]

(the notation here is compatible with J.E. Humphrey's "Introduction to Lie Algebras and Representation Theory") Let $\Phi \subset E$ be a root system. Let $\Delta \subset \Phi$ be a base. I already ...
9
votes
1answer
352 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 ...
9
votes
0answers
274 views

Abelian Cartan subalgebras

If a Lie algebra is semisimple or reductive, its Cartan subalgebras are Abelian, and their elements semisimple. Are there non-reductive algebras with Abelian Cartan subalgebras all of whose ...
8
votes
3answers
211 views

Representing $\mathbb{R}/\mathbb{Z}$ as a matrix group.

It was told to me that $G = \mathbb{R}/\mathbb{Z}$ is a real matrix group. Can someone help me understand how to represent $G$ in $Gl_n(\mathbb{R})$ for some $n$? (Supposedly, $n = 1$? But that's ...
8
votes
5answers
213 views

Getting started with Lie Groups

I am looking for some material (e.g. references, books, notes) to get started with Lie Groups and Lie Algebra. My motivation is that I (eventually) want to understand the theory underpinning papers ...
8
votes
2answers
365 views

Significance of the following Matrix?

I am unfamiliar with advanced Matrix theory (nor am I a mathematician), so please bear with me. Is there anything significant about the following Matrix structure? Are there any special symmetries or ...
8
votes
4answers
327 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty ...
8
votes
1answer
374 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 ...
8
votes
1answer
190 views

Schur -Weyl duality for $sl_2$ and $S_n$

$V$ is an $m$ dimensional vector space having a structure of $sl_2(\mathbb{C})$-module, where $sl_2(\mathbb{C})$ is the Lie algebra of the Lie group $SL_2(\mathbb{C})$. The symmetric group $S_n$ acts ...
8
votes
1answer
201 views

Ties between Lie algebras and ring theory

I would like to get a general understanding of the relationship between (noncommutative) ring theory and Lie algebra theory. All Lie algebras are finite dimensional and over a field $k$ of ...
8
votes
1answer
297 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, ...
8
votes
1answer
240 views

Proving that there exists a saturated set with given highest weight

This is an question about an exercise in Humphreys book on Lie algebras. First of all a bunch of definitions and notation, see §13 in Humphreys for details. Let $\Phi$ be a root system, $\Delta$ a ...
8
votes
1answer
201 views

Why is the Lie derivative linear in the vector field?

This might seem a very basic question, but I can't manage to find a proper proof in the books I have on my desk (or simply cannot see that it's "just that"). So be sure of what we talk about, let $G$ ...
8
votes
1answer
132 views

Trivial summand of a representation's symmetric power

The following comes from Exercise 13.17 of Fulton and Harris's book, Representation Theory: A First Course. Let $V$ denote the standard representation of $\mathfrak{sl}_3\mathbb{C}$, with weights ...
8
votes
0answers
107 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
8
votes
0answers
116 views

Why are parabolic subgroups called “parabolic” subgroups?

I used to think that things called "parabolic" must have something to do with parabolas or their defining quadratic equations. In fact, terms like parabolic coordinate, parabolic partial differential ...
7
votes
2answers
225 views

Are there finite-dimensional Lie algebras which are not defined over the integers?

Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra and let $R \subset \mathbb{C}$ be a subring. Say that $\mathfrak{g}$ is defined over $R$ if there exists a basis $x_1, ... x_n$ for ...
7
votes
1answer
648 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 ...
7
votes
1answer
197 views

Moment map of the action of $\operatorname{SO}(3)$ on the sphere

The moment map of the action of $\operatorname{SO}(3)$ on the sphere can be thought of as inclusion from $S^2$ into $\mathbb R^3$ by identifying $\mathfrak{so}(3)$ (the Lie algebra of ...
7
votes
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 ...
7
votes
1answer
299 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 ...
7
votes
2answers
324 views

Which Lie groups have Lie algebras admitting an Ad-invariant inner product?

I am trying to answer the following question: Which Lie groups have a Lie algebra admitting an $\text{Ad}$-invariant inner product? First of all, all compact Lie groups satisfy this condition ...
7
votes
1answer
132 views

Are Lie algebras of non-isomorphic central simple algebras non-isomorphic?

Comments to my answer to this MO question, which is isomorphic to this MSE question, point out that I was tacitly assuming that the associated Lie algebras to non-isomorphic quaternion algebras over ...
7
votes
1answer
195 views

Three-dimensional simple Lie algebras over the rationals

Let $\mathfrak g$ be a three-dimensional $\mathbf Q$-vectorspace endowed with the structure of a simple Lie algebra. How many non-isomorphic such $\mathfrak g$ are there? Over the complex numbers, ...
7
votes
1answer
153 views

Simplicity of $\operatorname{Der} \left(\mathbb F_p [x_1, \dots, x_n]/ (x_1^p,\dots, x_n^p )\right)$

I need to prove that the Lie algebra defined as: $W_{n} = \operatorname{Der} \left(\mathbb F_p [x_1, \dots, x_n ] / (x_1^p, \dots, x_n^p )\right)$, when $(x_1^p, \dots, x_n^p )$ is the ideal generated ...
7
votes
1answer
208 views

Isomorphism of an irreducible module of a certain Lie-algebra

While preparing for a test I found the next question which i cannot fully answer: Assume $k$ is an algebraically closed field, and $g_{1},g_{2}$ are $k$-Lie algebras and let $g=g_{1}\times g_{2}$. ...
7
votes
0answers
74 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 ...
7
votes
1answer
235 views

Dynkin diagram automorphisms and weights

Let $\sigma$ be a nontrivial Dynkin diagram automorphism of a finite-dimensional complex simple Lie algebra $\frak g$ (of type A, D or E) and let $\frak h$ be a Cartan subalgebra of $\frak g$. Let $I$ ...
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 ...
6
votes
3answers
435 views

Lie algebras and infinitesimals

I have seen at many places the notions that Lie Algebras are infinitesimal objects and they look really close at a point. But I never understood this. They are abstract algebraic objects different ...
6
votes
2answers
114 views

Equivalence of Two Lorentz Groups

How can I prove that $O(3;1)$ and $O(1;3)$ are the same group?