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
1answer
13 views

algebra operations

If $[x]=x^{i}\sigma^{i}$, Find an alternative form for the product $$ x^{i}n^{j}\sigma^{k}\epsilon^{ijk}=x^{i}(\vec{n} \times \vec{\sigma})^{i} $$ that has to be more compact than $$ ...
1
vote
0answers
45 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
0
votes
0answers
17 views

Criterion for semi-simplicity of Lie algebra generated by vector fields

Suppose I have a finite collection of smooth vector fields $V:=\{V_1,...,V_k\}$ on a smooth manifold $M$. Moreover suppose that the Lie algebra $g$, generated by $V$ (where the Lie bracket is defined ...
1
vote
1answer
41 views

Lie Algebra of a connected simple linear algebraic group

Let $G$ be a linear algebraic group and $A=K[G]$ (K is a field of characterstic 0) be the coordinate ring of $G$. In Humphreys, the Lie algebra of $G$ is defined as the space of left invariant ...
1
vote
0answers
52 views

Lie Algebra SU(2)

Given a two dimensional Hilbert-space, $\mathcal{H}$, and a vector $\eta \in \mathcal{H}$, of this space, if $\eta$ transforms in SU(2) like this, $$\eta \rightarrow e^{(-i\alpha ...
0
votes
1answer
48 views

Lack of associativity of cross product vs associativity of the exterior product

Can someone remind me in a nutshell why the associativity of the exterior product fails to transfer to the cross product? (It's been over a decade since I had to deal with the former back in school.) ...
0
votes
1answer
27 views

Cartan Lie Algebra of the Unitary Group $U(N)$?

I am having trouble understanding the Lie Algebra terminology. What is the Cartain Lie algebra of the unitary group $U(n)$? It must be in many textbooks, but they explain it very generally in terms ...
0
votes
1answer
51 views

Finding Lie algebra isomorphisms

I stumbled across exercises asking to prove the following isomorphisms: $\mathfrak{sl}_2(\mathbb{R}) \cong \mathfrak{so}_{2,1}(\mathbb{R})$ $\mathfrak{sl}_2(\mathbb{C}) \cong ...
0
votes
2answers
51 views

Is Cartan subalgebra of Complex semisimple Lie algebra the maximal Abelian subalgebra? I found two places give the different answers.

In wiki https://en.wikipedia.org/wiki/Cartan_subalgebra Example 4, it says that Cartan subalgebra of complex semisimple Lie algebra is not maximal Abelian subalgebra. However in Brian C. Hall's ...
1
vote
1answer
29 views

even orthogonal Lie algebra D2

Let $gl_S(n,F):= \{x ∈ gl(n, F) : x^tS = −Sx\}$. If $S=\begin{bmatrix} 0 & 1_l \\ 1_l & 0 \end{bmatrix}$, denote $D_l:=gl_S(n,F)$ with respect to above $S$ i.e the even orthogonal ...
1
vote
1answer
37 views

Special linear group

For the Lie algebra $sl(n,F)$, why $[sl(n, F), sl(n, F)] = sl(n, F)$ and why it's not true when n = 2 and $char F = 2$?
1
vote
0answers
24 views

showing that $\mathfrak{su}(n)$ is the only Lie subalgebra of $\mathfrak{u}(n)$ of dimension $n^2-1$

I came across the statement that for $\mathfrak{g}$ a Lie subalgebra of $\mathfrak{u}(n)$, $\text{dim}(\mathfrak{g})=n^2-1$ implies $\mathfrak{g}=\mathfrak{su}(n)$. I've tried the following. Does it ...
2
votes
0answers
26 views

Taylor series identity for polynomial using Lie group

The following question is from Kirillov's Introduction to Lie Groups and Lie Algebras, and my attempt is the following: $$\sum_{n\geq ...
11
votes
0answers
102 views

Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
2
votes
0answers
38 views

Why are $\mathfrak{pgl}_n\simeq\mathfrak{sl}_n$ when characteristic does not divide $n$?

Suppose $k$ is some algebraically closed field whose characteristic does not divide $n$. Why can we identify the lie algebras $\mathfrak{pgl}_n\simeq\mathfrak{sl_n}$ of the projective linear group and ...
2
votes
0answers
47 views

Standard Basis of $SU(2)$--where does the 1/2 come from?

The most common matrix representation of $SU(2)$ is given by $$ \begin{pmatrix} a & b\\ b^* & -a^*\\ \end{pmatrix} $$ where $a,b\in\mathbb{C}$. If we denote real components by the subscript ...
1
vote
0answers
23 views

Subalgebras of matrices property

Let $g$ be a Lie subalgebra of $gl_n(\mathbb{C})$ which has the propety that if $a\in g$ then also $a^\dagger\in g$ (where $a^\dagger$ is conjugate transpose). I want to show that if $a$ is an ideal ...
1
vote
1answer
48 views

Representation theory of Lie groups and outer automorphisms

If $G$ is a simply connected Lie group (I have in mind $G=SL_n(\mathbb{C})$), then we have an isomorphism $Aut(G)/Inn(G)\rightarrow Aut(g)/Ad(G)$ induced by taking the differential at $1$; here $g$ is ...
1
vote
0answers
14 views

$\mathrm{Rad}(L)$ is contained in all maximal solvable subalgebra. [duplicate]

Let $L$ be a Lie algebra and $\mathrm{Rad}(L)$ its unique maximal solvable ideal. Problem: Show that if $B$ is a maximal solvable subalgebra of $L$ (i.e. a Borel subalgebra) then ...
2
votes
1answer
78 views

Uniqueness of the killing form

I would like to consider/prove the following problems: let $k$ be a field, $g$ a finite-dimensional simple Lie algebra over $k$ with Killing form $B$. If $\sigma:g\times g\rightarrow k$ is a ...
1
vote
0answers
16 views

left and right jacobians (Not derivatives) of a Lie group

Let $G<Gl(n)$ be a Lie group, and let $g:\mathbb{R}^n\rightarrow G$ be a smooth curve, parametrized as $g(q(t))$ with $t\in\mathbb{R}$. I understand that in that case it holds that $$ \tag 1 ...
0
votes
0answers
50 views

Lie algebra of a finite group

I'm trying to find the normalizer of the Pauli group $G_n$ (as a subgroup of $SU(2^n)$) utilizing Lie algebras, as is done in a reference to find the normalizer of the Heisenberg group $HW(n)$. There, ...
0
votes
0answers
27 views

Special linear Lie algebra.

When I am reading the Notes on Lie algebra by Hans Samelson, there is a sentence: The standard skew-symmetric (exterior) form $det[X, Y ] = x_1y_2−x_2y_1$ on $\mathbb{C}^2$ is invariant under ...
2
votes
2answers
73 views

Why, for nilpotent Lie Algebras, is the inclusion to the derivations $x \mapsto {ad}_x$ not surjective?

Let $\mathfrak g$ be a finite dimensional nilpotent Lie Algebra over a field $F$ (is characteristic zero necessary?). Why is the map $ad: \mathfrak g \to Der (\mathfrak g), x \mapsto {ad}_x$ not ...
0
votes
0answers
17 views

sl_2 triple and comparing two nilpotent orbits in a Lie algebra

I've been working on a question on local Galois deformation theory. It eventually boils down to the following questions on nilpotent elements in a complex semisimple Lie algebra $\mathfrak g$ which is ...
6
votes
0answers
80 views

A subspace is invariant by the Lie group if it is invariant by the Lie algebra

Let $G$ be a connected Lie group and $$\varphi:G\to \mathrm{GL}(V)$$ a representation on a finite dimensional real vector space $V$. Let $$\psi:\mathfrak{g}\to\mathrm{End}(V)$$ be the associated Lie ...
1
vote
0answers
27 views

Commutator of Lie sub-algebra

I have a problem understanding the proof of Proposition 8.20, page 211, in Besse's Einstein Manifolds. He considers a semi-simple Lie algebra $\mathfrak{g}$ with $\operatorname{Ad}$-invariant scalar ...
2
votes
1answer
28 views

Push forward of the Lebesgue measure is the Haar measure of the Carnot group

I have the following problem. I have a Carnot group $(\mathbb G,*)$ which is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ is stratified as $\mathfrak g= ...
1
vote
1answer
78 views

When considering vector spaces $A$ and $B$, what is the difference between $A \times B,$ $A \otimes B$ and $A \wedge B?$

I have looked at this resource http://hitoshi.berkeley.edu/221a/tensorproduct.pdf to instinctively differentiate between the tensor product and the direct sum of two vector spaces. I am currently ...
2
votes
1answer
64 views

How to show $\operatorname{Out}(\operatorname{SL}_3(\mathbb{C})) \cong \mathbb{Z}/2\mathbb{Z}$?

Let $G$ be a Lie group, $\operatorname{Aut}(G)$ the group of diffeomorphisms of $G$ that are also homomorphisms. Denote by $\operatorname{Inn}(G) \unlhd \operatorname{Aut}(G)$ the group of ...
1
vote
1answer
31 views

What is a Hamiltonian in a Poisson algebra?

Classical physics on the phase space $T^* M$ (with $M$ a smooth manifold) is done mostly in the following way: one endows $T^*M$ with a Riemannian structure $g^*$ (that will give the kinetic term) and ...
1
vote
2answers
114 views

$\mathfrak{sl}_2(\mathbb{R})$ and $\mathbb{R}^3$ as subalgebras of $\mathfrak{sl}_2(\mathbb{C})$

I have an exercise which asks to prove that the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ contains the real, non-isomorphic subalgebras $\mathfrak{sl}_2(\mathbb{R})$ and $\mathbb{R}^3$ and to show ...
0
votes
0answers
34 views

Lie algebras over non-algebraically closed field

Could someone recommend a book on Lie algebras over a not-necessarily algebraically closed field? I am particularly interested in the representation theory, so it should contain all the usual results ...
0
votes
0answers
10 views

Arrive to the group law in exponential coordinates using the vector fields expressed in exponential coordinates

I need an help with the following question. I have this definition for Engels group $\mathbb E$: it is the only connected and simply connected Lie group that has the Engels algebra $\mathfrak g$ as ...
0
votes
1answer
56 views

Constructive proof that the Lie functor is faithful?

I am wondering how to show that the Lie functor taking Lie groups to Lie algebras is faithful? In particular, I am looking for a constructive proof, since I am working in the context of synthetic ...
0
votes
1answer
70 views

How to show that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple?

We defined a Lie algebra to be simple, if it has no proper Lie ideals and is not $k$ (the ground field). We have the proposition that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ ...
1
vote
0answers
41 views

Basis for traceless, symmetric matrices?

Consider, for example, the set of of all symmetric, traceless $4 \times 4$ matrices. I'm trying to find a correctly normalized basis for this set. So far, I have $$s(1)=\left( \begin{array}{cccc} 0 ...
3
votes
1answer
56 views

Examples of Induced Representations of Lie Algebras

Given a (finite-dimensional) Lie algebra $\mathfrak{g}$, a subalgebra $\mathfrak{h}\subset\mathfrak{g}$, and a representation $\rho:\mathfrak{h}\rightarrow\mathfrak{gl}(V)$ of $\mathfrak{h}$, one can ...
3
votes
0answers
83 views

Why do we need finite dimension for this diagram to commute?

Let $\frak g$ be a Lie algebra and $\rho$ be a finite dimensional representation of $\frak g$ in a linear space $V$. They define the momentum associated to $\rho$ as the map $\mu: V \otimes V^\ast \to ...
0
votes
1answer
28 views

Treatments of Lie theory and Noether theorems in tensor notation

I am looking for a conceptual treatment of Lie theory and Noether theorems that uses tensors calculus rather than exterior calculus. I know that tensor calculus is not optimal for these subjects, ...
0
votes
2answers
67 views

Relating ${\rm ad}(\phi(X))$ with $\phi \circ\,{\rm ad}(X) \circ \phi^{-1}$.

The exercise is to prove that $\phi: {\frak g}\to {\frak g}$ is an automorphism if and only if ${\rm ad}(\phi(X)) = \phi\circ {\rm ad}(X)\circ \phi^{-1}$. Here $\frak g$ is a Lie algebra, of course. ...
1
vote
0answers
42 views

The condition for the exponential map of Lie group is surjective or injective

$G$ is a connected Lie group, $g$ is its Lie algebra. 1) What is the necessary and sufficient condition for the exponential map from $g$ to $G$ is surjective? 2) What is the necessary and ...
0
votes
1answer
13 views

Normal subgroup of Engel group

The Engel algebra $\mathfrak g$ is the Lie algebra generated, as a vector space, by four vectors $X_1,X_2,X_3,X_4$ with the only non trivial commutation relations:$$[X_1,X_2]=X_3, \quad ...
0
votes
0answers
42 views

Advantages and disadvantages of defining Lie bracket via right invariant vs. left invariant vector fields

I was just wondering what are the advantages and disadvantages of the two conventions used for defining Lie brackets? For example, if we use right invariant vector fields as the convention for ...
4
votes
2answers
64 views

One-parameter subgroups and Lie bracket

Suppose that $G$ is a Lie group. It is easy to prove that given $X \in Lie(G)$ there exists a unique one-parameter subgroup $\phi_X : \mathbb{R} \to G$ such that $\dot{\phi}(0)=X$. My question is: ...
2
votes
0answers
47 views

The simplest way to present a Lie algebra to a wide audience?

I would like to get suggestions from you as to the best way to present the idea and contents of Lie algebras to a wide public of people with no detailed background in maths. What wlould you explain to ...
0
votes
0answers
21 views

Adjoint action on Lie algebra su(2) ($A \in SU(2), X \in \mathfrak{su(2)} \Rightarrow AXA^{-1}\in \mathfrak{su(2)}$)

I am trying to understand ho $SU(2)/\{\pm I\} \cong SO(3)$ (see: how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$) but i am not sure about the adjoint action. In especially, as I understand, the adjoint ...
1
vote
1answer
45 views

$su(2) $ and $ sl(2;R)$ are not isomorphic? [duplicate]

As real Lie algebras, both are three-dimensional. The basis of $su(2)$ is $$ \left( \begin{matrix} i & 0 \\ 0 & -i \end{matrix} \right), \left( \begin{matrix} 0 & 1 \\ -1 & 0 ...
0
votes
0answers
28 views

Lie functor produces an antihomomorphism in Lavendhomme's synthetic differential geometry text?

Classically the Lie functor maps a Lie group homomorphism to a Lie algebra homomorphism. But in Proposition 15 on page 249 in Basic Concepts of Synthetic Differential Geometry, Lavendhomme states that ...
2
votes
2answers
65 views

Lie algebra of the automorphism group of a Lie group?

Let $G$ be a Lie group and $\text{Aut}(G)$ the group of all Lie group automorphisms of $G$. If $\text{Aut}(G)$ can be interpreted to be a Lie group (for example, in the context of synthetic ...