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)

0
votes
1answer
57 views

Question about Lie algebra, where am I going wrong?

Let $M:= \begin{pmatrix}1 & 2 & 3 & 4\\3 & 4 & 4 & 1\\1 & 9 & -1 & -3\\9 &5 & -2 & -4\end{pmatrix} = \begin{pmatrix} A & B\\C & ...
1
vote
1answer
135 views

Conjugate exponential integral formula for Lie algebra

Somewhere in my notes the following formula appears $\int_0^1 e^{s R} \frac{d R}{dt} e^{(1-s)R} ds = \frac{d}{dt} e^{R}$ where $R$ depends on $t$, and has values in a Lie algebra [$\mathfrak{so}(3)$ ...
2
votes
0answers
125 views

Finding the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$

How can I find the Killing form of $\mathfrak{sp}_{2n}(\mathbb{C})$? I'm explicitly working with basis vectors in trying to compute $\operatorname{tr}(\operatorname{ad}(a)\operatorname{ad}(b))$ but ...
1
vote
0answers
66 views

Question about root space

Let $\mathfrak{g}$ be a Lie algebra and consider $\operatorname{Rad}(\mathfrak{g})$, the radical of $\mathfrak{g}$, that is, the sum of all solvable ideals in $\mathfrak{g}$. Suppose that we have the ...
1
vote
1answer
72 views

Understanding a statement about the existence of functionals

Thm. Let $L$ be a solvable subalgebra of $gl(V)$, $V$ a finite dimensional nonzero vector space. Then $V$ contains a common eigenvector for all the endomorphisms in $L$. The proof of this theorem is ...
3
votes
1answer
101 views

Lie brackets of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$

Let $f$ be a smooth function, $f\colon\mathbb{R}^2\to \mathbb{R}$. What is $\left[\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right]$ ? I want to say it's $0$ since $\frac{\partial ...
5
votes
1answer
635 views

Computation of the Killing form of $\mathfrak{gl}_{m}$.

Consider the Killing form of the Lie algebra $\mathfrak{gl}_{m}$. Then $\{e_{ij}\}$ is a basis for this Lie algebra where $e_{ij}$ is a matrix with 1 in the $i$th row, $j$th column and 0 everywhere ...
3
votes
3answers
352 views

Is Cartan subalgebra of a Lie algebra unique?

Let $\mathfrak{g}$ be a Lie algebra. Is Cartan subalgebra of $\mathfrak{g}$ unique? I see in some places it is written "Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$".
4
votes
0answers
104 views

The Nambu bracket

Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $$\{F,G\}_c = \langle\nabla ...
0
votes
2answers
97 views

Can you cancel in characteristic p?

Lets say you have an ideal in some algebra of characteristic p. Yeah, so if you have a lie algebra with a field that is characteristic p. Can you cancel. So for example if you have a vector space ...
1
vote
1answer
230 views

Showing Witt Algebra is simple?

So I got the Witt Algebra over finite field http://en.wikipedia.org/wiki/Witt_algebra and need to show that it's simple. But, I don't know where to start. So let I be a nonzero ideal of witt algebra, ...
1
vote
1answer
78 views

algebraic group to the lie algebra and hom

Let $G$ be a linear algebraic group and let $\rho:G \rightarrow GL(V_{1})$ and $\psi:G \rightarrow GL(V_{2})$ be finite representations. Why is $Hom_{G}(V_{1},V_{2}) \subset Hom_{\mathfrak ...
2
votes
0answers
65 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
0
votes
1answer
178 views

bilinear forms and linear algebraic groups

Let $G$ be a linear algebraic group and $\phi$ a finite dimensional regular representation of $G$ into $GL(V)$ I would like to know about bilinear forms on $V$ and when they are $G$-invariant. ...
8
votes
2answers
750 views

Universal Cover of $SL_{2}(\mathbb{R})$

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
3
votes
1answer
140 views

What is the difference between $\ker( L \bigwedge L \overset{[-,-]}{\rightarrow} L )$ and $\langle a \wedge b \big| [a,b]=0\rangle$?

Let $L$ be a finite dimensional Lie algebra. We view the Lie bracket as a linear map on the exterior square: $$\pi:L \bigwedge L \rightarrow L$$ Define $$\bigwedge L := \langle a \wedge b \big| ...
2
votes
2answers
247 views

Is there some kind of character theory for representations of finite dimensional algebras?

We know that for a representation $V$ of a Lie algebra or a quantum group, we can define character of $V$ as $ch(V)=\sum_{\mu} dim(V_{\mu})e^{\mu}$, where $V_{\mu}$ is the weight space of $V$ with ...
1
vote
1answer
1k views

Lie derivative of a vector field equals the lie bracket

Let $X$ and $Y$ be vector fields on a smooth manifold $M$, and let $\phi_t$ be the flow of $X$, i.e. $\frac{d}{dt} \phi_t(p) = X_p$. I am trying to prove the following formula: $\frac{d}{dt} ...
0
votes
1answer
255 views

Lie algebra using skew-symmetry

Let g be a Lie algebra such that [[x,y],y]=0 for all $x,y \in g$. Show that 3[[x,y],z]=0 for all $x,y,z \in g$. [Hint: Observe that the mapping (x,y,z) to [[x,y],z] is skew-symmetric in x,y,z and make ...
2
votes
1answer
281 views

Lie algebra is associative

Prove that a Lie algebra g is associative iff the derived subalgebra of g is contained in the centre of g, that is $g^{(1)} \subset c(g)$. So we have the derived sub algebra is in the centre of g. We ...
1
vote
1answer
419 views

Derivation of a lie algebra

Let A be an algebra over K with multiplication $(x,y) \rightarrow x \cdot y$. A linear operator D on the vector space A is called a derivation of A if $D(x \cdot y)=(Dx) \cdot y + x \cdot (Dy)$ $( ...
3
votes
1answer
125 views

Adjoint endomorphism on $\mathfrak{sl}(2,k)$

Let $\{e,h,f\}$ be the standard basis of the Lie algebra $\mathfrak{sl}(2,k)$. Prove that $(\mbox{ad }e)^3=0$ http://en.wikipedia.org/wiki/Special_linear_Lie_algebra First I computed $(\mbox{ad ...
7
votes
4answers
635 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
2
votes
1answer
153 views

Computing the Lie algebra of $SU(n, \mathbb{C})$

The group $SU(n, \mathbb{C})$ is the set of $n \times n$ complex matrices $Q$ such that $\det Q = 1$ and $Q\overline{Q}^{T} = 1$. The Lie algebra is the set of traceless anti-Hermitian matrices. To ...
6
votes
1answer
140 views

Kernel of the Lie bracket

Let $\mathfrak{g}$ be a dimension 3 Lie algebra and $[\quad,\quad]$ be a rank 1 map from $\bigwedge^{2}\mathfrak{g} \rightarrow \mathfrak{g}$. In this case, the kernel of $[\quad,\quad]$ is $3 - 1 = ...
8
votes
4answers
348 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 ...
3
votes
1answer
178 views

Prove that the weight space of the highest weight is always one dimensional for an irreducible representation.

Prove that the weight space of the highest weight is always one dimensional for an irreducible representation, where the weight has the usual meaning in representation theory. If H is a operator and ...
1
vote
1answer
133 views

Basic Lie Algebra Question

Essentially, I'm trying to prove that when computing the tangent space for a group that there's nothing special about considering it at only the identity. Namely, there is an isomorphism of vector ...
0
votes
1answer
110 views

Question about the abelianization of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional Lie algebra and $\mathfrak{h}$ a Lie subalgebra. If I know that $\mathfrak{g}/\mathfrak{h}$ is abelian, does it follow that $\mathfrak{g}/\mathfrak{h} ...
2
votes
0answers
111 views

Question about a Corollary of Engel's Theorem

Engel's Theorem states that: Let $L$ be a subalgebra of $\mathfrak{gl}(V)$, $V$ finite dimensional. If $L$ consists of nilpotent endomorphisms and $V \neq 0$, then there exists nonzero $v \in V$ for ...
2
votes
2answers
322 views

Schur's Lemma for algebraically closed field .

What is Schur's Lemma ? and why is it valid only for the algebraically closed field ?
4
votes
1answer
255 views

Lie algebra associated with the orthogonal group $\operatorname{SO}(2n)$?

How do I prove that the lie algebra associated with the even dimensional orthogonal group $\operatorname{SO}(2n)$ is given by matrices $B$ satisfying $B^\top K + KB = 0$, where $K = U^\top U$, $U$ ...
3
votes
1answer
970 views

Lie algebra associated with the symplectic group Sp(2n)?

The group Sp(2m) consists of the 2m×2m matrices A with the property At (A transpose)JA = J, where J is the 2mX2m standard skew-symmetric matrix. How do I prove that the lie algebra associated with has ...
1
vote
0answers
192 views

$SU(2)$ is a covering space of $SO(3)$.

The method of topology is very clear.Then there's a question asking to use adjoint representation of lie group $SU(2)$ $(\operatorname{adj}:SU(2)\to GL(su(2)))$to prove this. I can't solve this .
4
votes
2answers
147 views

Is the multiplicative complex plane a Lie group?

I know that the complex plane is a Lie group with +, but is it also a Lie group with the usual complex multiplication? This would give us a nice geometrical interpretation of the famous Euler ...
7
votes
1answer
301 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 $I$ ...
1
vote
1answer
185 views

question on roots and root vectors of a simple lie algebra

Assuming that for each root α there is only one linearly independent root vector, show that if $\alpha$, $\beta$, and $\alpha+\beta$ are roots, then [$e_\alpha$ , $e_\beta$ ] not equal to 0. Here ...
1
vote
1answer
84 views

If $\alpha$ is a root of a simple lie algebra, then prove that the only multiples of $\alpha$ which are roots are $\alpha, -\alpha,0$

If $\alpha$ is a root, then the only multiples of α which are roots are $\alpha, -\alpha, 0$. Here $\alpha$ is a root of a simple lie algebra. How do I prove this?
1
vote
1answer
64 views

If $\alpha$ is a root of a simple lie algebra, prove that $\langle \alpha,\alpha \rangle \neq 0$

If $\alpha$ is a root of a simple lie algebra, prove that $\langle \alpha,\alpha \rangle$ not equal to $0$. From this, I want to prove that the $\langle,\rangle$ could be used as a scalar product.
3
votes
1answer
280 views

Conditions for left-invariant one-forms to be closed.

Let $G$ be a connected (semisimple) Lie group with Lie algebra $\frak{g}$. For $\omega \in \frak{g}^*$, we may define a left invariant one-form on $G$ by $\left[ \omega (g)\right] (v)=\omega \left( ...
3
votes
1answer
137 views

How do I find all the roots of a lie algebra and hence its root system diagram given the cartan matrix for that algebra?

If I am given the cartan matrix, I can find the $2\langle αi , αj\rangle/\langle αj , αj\rangle$ of the simple roots where $α$ are the simple roots. But, from this how do I find the $\langle αi , ...
1
vote
0answers
54 views

$G_1$-Scalar factors for Clebsch-Gordan coefficients for $ U(n)$

when evaluating the $G_1$ scalar factors for CGC's of $U(n)$ it seems that some of the factors are undefined. The explicit formula for the evaluation of the scalar factors is Eq. (6) in 18.2.8 of N.J. ...
1
vote
1answer
88 views

Question on induction proof (about direct sums of irreducible submodules)

Let $V$ be an $L$-module. I want to show that $V$ is a direct sum of irreducible $L$-submodules if each $L$-submodule of $V$ possesses a complement. I want to show this via induction on the dimension ...
2
votes
2answers
269 views

Proving a Lie algebra is simple

Let L be a 3-dimensional vector space over k with basis x,y,z. Given L an anti-commutative algebra structure by setting $[x,y]=z,[y,z]=x,[z,x]=y$ Prove that L is a simple Lie algebra. So L is ...
5
votes
1answer
438 views

How to prove that a subalgebra $\mathfrak p$ in a semisimple lie algebra $\mathfrak g$ is parabolic if $\mathfrak p^\perp=\mathfrak{rn}(\mathfrak p)$?

Let $\mathfrak g$ be a semissimple Lie algebra over an algebraically closed field of characteristic zero. Suppose $\mathfrak p$ is a subalgebra of $\mathfrak g$ such that $\mathfrak ...
2
votes
0answers
77 views

Principal Bundles and Lie algebras

Suppose I have a principal $S^{1}$ bundle over a nice compact symmetric space. The symmetric space arises as a homogeneous space, call it $X=G/H$. On the Lie algebra level we have the decomposition ...
19
votes
5answers
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 ...
0
votes
1answer
672 views

Basis for adjoint representation of $sl(2,F)$

Consider the lie algebra $sl(2,F)$ with standard basis $x=\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$, $j=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $h=\begin{bmatrix} 1 & 0 ...
2
votes
2answers
43 views

Showing $R(G) = R(T)^W$

Let $G$ be a compact connected Lie group and $T$ a maximal torus. Let $R(G)$ be the representation ring of $G$. Then restriction of reps gives a map $R(G) \to R(T)^W$, where $R(T)^W$ are the ...
1
vote
0answers
125 views

First-order derivatives in differential forms calculus

Let $d$ denote the Cartan differential, and let $\delta$ denote the codifferential. The underlying domain is not important for what follows. The canonical generalization of the Laplace-operator ...