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
3answers
132 views

Question about divisibility of the factorial

I've found this question in the book devoted to Lie's groups. How to prove that $n!$ is devided by $n_{1}!n_{2}!...n_{k}!$, where $n=n_{1}+n_{2}+...+n_{k}$ and $n$, $n_{i}$ are natural numbers?
3
votes
0answers
139 views

The enveloping algebra of a finite dimensional Lie algebra has no zero divisor

Let $L$ be a complex, finite dimensional Lie algebra. It is well-known that the graded associative algebra of the enveloping algebra $U(L)$ is isomorphic to the symmetric algebra $S(L)$. Therefore ...
4
votes
0answers
154 views

Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$

I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189. Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
9
votes
2answers
562 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 ...
0
votes
1answer
66 views

Prove that $\frac{q(r+1)}{(\beta,\beta)}=\frac{q'(r'+1)}{(\alpha,\alpha)}$

Let $\alpha $,$ \beta$ $\in$ $\Phi$. Let the $\alpha$-string through $\beta$ be $\beta-r\alpha$,$\ldots$,$\beta+q\alpha$,and let the $\beta$-string through $\alpha$ be $\alpha-r'\beta$, ...
5
votes
1answer
283 views

Irreducibility and weights of a representation

For some reason I can't get a good hold of those topics (I'm reading Brian C. Hall's Lie Groups, Lie algebras and Representations. So it's matrices only). I'll try to narrow it a bit more: ...
2
votes
1answer
126 views

Complexifying a group action of SL(n, R) to a group action of SL(n, C)

Given an analytic group action of $SL(n, \mathbb{R})$ on $\mathbb{R}^m$ fixing the origin, in this article the author then proceeds to "complexify the analytic $SL(n, \mathbb{R})$ action to obtain a ...
5
votes
2answers
172 views

Understanding an Example of a Lie Groupoid

For the definition of Lie groupoid see https://en.wikipedia.org/wiki/Lie_groupoid . In this question I want to understand Example 1.1.17 in "General theory of Lie groupoid and Lie algebroids" by ...
7
votes
1answer
537 views

Reference for Lie-algebra valued differential forms

I am learning about vector-valued differential forms, including forms taking values in a Lie algebra. On Wikipedia there is some explanation about these Lie algebra-valued forms, including the ...
0
votes
0answers
41 views

The simplicity of $\bigwedge^i \mathbb{C}^{n+1}$ as a representation of $\mathfrak{sl_{n+1}}$ and its weight vectors

I want to show that $\bigwedge^i \mathbb{C}^{n+1}$ is a simple representation for $\mathfrak{sl}(n+1,\mathbb{C})$ for each $1\le i \le n+1$ but I'm already stuck at determining the weight vectors. So ...
1
vote
1answer
79 views

Representations of semisimple Lie algebra

Let $L$ a Lie algebra and $V$ a representation of $L$. We define $$ V^{L}:= \{ v \in V \, | \, xv=0, \, \forall x \in L \} $$ and $V_{L}:=V/LV$. Let $\pi: L \rightarrow V_{L}$ be the quotient ...
0
votes
1answer
69 views

Irreducible action Lie algebras

How can I show that the the action of Lie algebra $\mathfrak{so}(2n+1)$ on $\mathbb{C}^{2n+1}$ is irreducible? Is there a simple way?
1
vote
1answer
83 views

Invariant form on Lie algebra

Does anyone have a reference for the following fact? Let $G \subset GL(n)$ be a compact Lie group. Then the form $$f(A)=-Tr(A^2)$$ defined for $A \in T_e G \subset \mathfrak{gl}(n)$ is positive ...
2
votes
0answers
24 views

Integer domain enveloping algebra

I must prove that if $L$ is a Lie algebra and denoting $U(L)$ the enveloping algebra, then $U(L)$ hasn't zero divisions (e.g. if $ab=0 \,\,\, a,b \in U(L)$ then $a=0$ or $b=0$). Some ideas?
3
votes
1answer
112 views

Lie algebras and roots systems

Let $\Phi$ an irreducible system of roots, $\Phi^{+} \subset \Phi$ a choose of positive roots. I have to prove that if $(\alpha, \beta) \ge 0$ for al $\beta \in \Phi^{+}$ then $\alpha$ is the highest ...
3
votes
1answer
119 views

Lie algebra and enveloping algebra

I have to prove that a Lie algebra over the field $k$ is trivial if and only if the enveloping algebra $U(L)=k$. I have an idea of proof: If $L=\{0\}$ we have that the tensor algebra $T^m=\{0\}$ for ...
1
vote
0answers
37 views

System of roots [duplicate]

Let $\Phi$ an irreducible system of roots, $\Phi^{+} \subset \Phi$ a choose of positive roots. I have to prove that if $(\alpha, \beta) \ge 0$ for al $\beta \in \Phi^{+}$ then $\alpha$ is the highest ...
9
votes
1answer
460 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 ...
3
votes
1answer
501 views

Conjugate Representations

Are there any general results on when conjugate representations of a real Lie algebra are equivalent? I'm inclined to say that they are often not, but this is merely going on my case by case ...
1
vote
1answer
71 views

When can you build up all representations from the fundamental and antifundamental ones?

Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I ...
3
votes
1answer
169 views

When is the Killing form null?

When is the Killing form $\kappa$ of a Lie algebra $\mathfrak g$ null, i.e. $\kappa(\cdot,\cdot)=0$? Surely this is true for any Lie algebra with trivial bracket, but is this the only case? I can't ...
4
votes
1answer
348 views

How can the Cartan-Weyl basis of su(2) be a basis if it does not consist of antihermitian operators?

Consider a Lie algebra. The ladder operators (i.e. root vectors, or eigenvectors of the Cartan subalgebra with respect to the adjoint representation) form a handy basis of the algebra called a ...
1
vote
0answers
129 views

Conjugate Representations for $\mathfrak{sl}(2,\mathbb{C})$

Let $\mathfrak{sl}(2,\mathbb{C})$ be the complex Lie algebra of $SL(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ be its realification; that is $\mathfrak{sl}(2,\mathbb{C})_\mathbb{R}$ ...
3
votes
2answers
62 views

Not null Killing form.

I have to find an example of solvable Lie algebra $L$ such that the Killing form of $L$ isn't null. If we take the Borel subalgebra of $\mathfrak{sl}(2)$, we have that the Killing form of $L$ is the ...
2
votes
1answer
208 views

Conjugate Representations of Lie Algebra of Lorentz Group

I'm trying to understand the Lie algebra of the Lorentz group and am almost there, but am stuck at the final hurdle! It's easy to prove that $$\frak ...
5
votes
2answers
309 views

Representations of Direct Sum of Lie Algebras

I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
3
votes
1answer
336 views

Tensor product Lie algebras

How can I show that $\mathfrak{so}(3, \mathbb{R}) \otimes \mathbb{C} \simeq \mathfrak{sl}(2)$? Is there a way that doesn't involve systems of roots?
2
votes
1answer
131 views

Isomorphism with Lie algebra $\mathfrak{sl}(2)$

Let $L$ be a Lie algebra on $\mathbb{R}$. We consider $L_{\mathbb{C}}:= L \otimes_{\mathbb{R}} \mathbb{C}$ with bracket operation $$ [x \otimes z, y \otimes w] = [x,y] \otimes zw $$ far all $x,y \in ...
1
vote
1answer
324 views

Lie Algebra of the Lorentz Group $SO(1,3)^{\uparrow}$

I'm trying to get my head around the Lie algebra of the Lorentz group once and for all, but have got tied up in knots. Where is my error in the following? The universal covering group of the Lorentz ...
6
votes
1answer
160 views

$\mathfrak{sl}(2,\mathbb C)$ real v. complex

I'm a Lie theory novice, so please bear with me. My understanding is that the Lie algebra $\mathfrak g$ of a matrix Lie group $G$ is the pair $(V, [\cdot, \cdot ])$ where $V$ is the real vector space ...
3
votes
3answers
314 views

Free Lie algebra.

Let $F$ be the free Lie algebra on $\{x,y,z\}$ and $L$ the quotient of $F$ by the ideal $I$ generated by brackets that involve at least three free generators. I have to prove that $dim(L)$ is $6$ and ...
2
votes
1answer
98 views

Complexificantion of a Lie algebra.

Let $L$ be a real Lie algebra. We can consider $L_{\mathbb{C}}:=L \otimes_{\mathbb{R}} \mathbb{C}$ the complexification of $L$. So on $L$ we can define the bracket operation: $[x \otimes z, y \otimes ...
0
votes
1answer
60 views

Enveloping algebra of a Lie algebra

Let $U(L)$ be the enveloping algebra of a Lie algebra $L$. How can I prove that $U(L)$ hasn't zero divisiors (e.g. if $xy=0$, $x,y \in U(L)$ then $x=0$ or $y=0$)?
2
votes
1answer
133 views

Nilpotent Lie algebra

I have to find an example of non nilpotent Lie algebra $L$ and an ideal $I$ of $L$ such that $L/I$ is nilpotent. So we can take the algebra of $ 2 \times2$ matrix upper triangular and with null trace. ...
5
votes
1answer
443 views

Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
1
vote
1answer
103 views

Semisimplicity, simplicity and type of symplectic algebra $sp(2n)$

Let $sp(2n)$ be the symplectic algebra. I have to prove that $sp(2n)$ is a simple algebra and its type is $C_{n}$. In order to prove the semisimplicity we can consider the this theorem: Let $V$ be a ...
4
votes
1answer
110 views

The set of complete vector fields

The set of all complete vector fields in $\mathbb R^{n}$ is closed under Lie bracket? is this set a $D$-module where $D$ is the ring of bounded smooth funcions? Can anyone recomend me a book on the ...
3
votes
1answer
150 views

Induced metric via $\mathbb C P^n \cong SU(n + 1)/S(U(n) + U(1))$

I was wondering if the homeomorphism above gives me the Fubini Study metric on $\mathbb C P^n$. More precisely: Consider $\mathbb C P^n$ equipped with the metric induced by the standard construction ...
1
vote
0answers
281 views

jacobian involving SO(3) exponential map: $\log(R \exp(m))$

I would like to compute the 3 × 3 Jacobian of $$ \log(R \exp(m)) $$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...
10
votes
2answers
2k 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 ...
2
votes
1answer
121 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
6
votes
2answers
361 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
0
votes
1answer
234 views

How to verify the Jacobi identity for the semidirect product Lie algebra

I've been trying to check the claim that the vector space direct sum $L \oplus D$ is a Lie algebra, and I'm having a lot of trouble with verifying the Jacobi identity. It's defined where $L$ is a Lie ...
2
votes
1answer
58 views

Representation of non-Abelian, dimension 2 Lie algebra

Let $k$ be a field and $\mathfrak{g}=kx\oplus ky$ with $[x,y]=y$. Show that $\rho(x)=t\,\frac{d}{dt}$ and $\rho(y)=t\cdot$ (mult. by $t$) define a representation $\rho:\mathfrak{g}\to ...
4
votes
1answer
120 views

Centre of $\mathfrak {sl}_{3}\mathbb{C} $ Lie algebra

$\mathfrak {sl}_{3}\mathbb{C}$ is the Lie algebra of $3\times3$ matrices with complex entries and trace $0$ and Lie bracket $[X,Y] = XY-YX \hspace{3mm} \forall \hspace{3mm} X,Y\in \mathfrak ...
3
votes
1answer
560 views

Lie algebra and Killing form

Let $L$ be a Lie algebra. I have to prove that if $L$ is a simple Lie algebra every bilinear associative form (e.g. $([x,y],z)= (x,[y,z])$ for all $x,y,z \in L$) is a multiple of Killing form.
1
vote
0answers
115 views

Root system of a Lie Algebra

Could anybody help me to solve this problem with roots system? Be $\Phi$ an irreducible root system. $\Phi^{+}$ a choice of positives roots in $\Phi$. Prove that if $(\alpha,\beta)\ge0$ $\forall ...
0
votes
1answer
73 views

Differences between a Cartan subalgebra and a Levi subalgebra?

Let $\mathfrak{h}$ be a Cartan subalgebra and $\mathfrak{l}$ be a Levi subalgebra of $\mathfrak{gl_n}$, where $\mathfrak{h}$ and $\mathfrak{l}$ are both semisimple subalgebras. This is a simple ...
4
votes
1answer
227 views

Computing Lie algebra homomorphism from Lie group homomorphism

I'm pretty much stuck on the following question (taken from the book Lie groups and introduction to linear groups by Rossman W.): I've found some clues, but I think I lack proper understanding of ...
1
vote
1answer
97 views

surjective Lie algebra homomorphism preserves center

If $\phi: L_1 \rightarrow L_2"$ is a surjective Lie algebra homomorphism, is it true that $\phi (Z(L_1))=Z(L_2)$. I see that $\phi (Z(L_1))$ is in $Z(L_2)$, but if $\phi^{-1}(0)$ is not $0$, i.e ...