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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72 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 ...
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56 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 ...
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1answer
158 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. ...
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2answers
562 views

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

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
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1answer
132 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| ...
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195 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 ...
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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} ...
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1answer
210 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 ...
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1answer
214 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 ...
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1answer
271 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 $D(x \cdot y)=(Dx) \cdot y + x \cdot (Dy)$ $( ...
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1answer
105 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 ...
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4answers
541 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 $\operatorname{Tr}$ ...
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1answer
140 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 ...
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1answer
134 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 = ...
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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 ...
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1answer
153 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 ...
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105 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 ...
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1answer
85 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} ...
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101 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 ...
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2answers
258 views

Schur's Lemma for algebraically closed field .

What is Schur's Lemma ? and why is it valid only for the algebraically closed field ?
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1answer
199 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 $K = U^\top U$, $U$ ...
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178 views

How to prove the lie algebra of $n\times n$ traceless matrices is semi-simple?

The Lie algebra of all the $n \times n$ matrices is not semi-simple. However, if we restrict ourselves to traceless $n\times n$ matrices, we do obtain a semi-simple (in fact, simple) Lie algebra which ...
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1answer
667 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 ...
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183 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 .
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146 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 ...
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1answer
234 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 $I$ ...
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1answer
160 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 ...
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1answer
80 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?
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1answer
53 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.
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1answer
204 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( ...
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1answer
131 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 , ...
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0answers
43 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. ...
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1answer
86 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 ...
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213 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 ...
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1answer
365 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 ...
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70 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 ...
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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 ...
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1answer
465 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 ...
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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 ...
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120 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 ...
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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 ...
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32 views

Understanding the Lie algebra $o_{V,B}$

I am learning about Lie algebras and I do not understand the following subalgebra of $\mathfrak gl_{V}$. Let $V$ be a vector space and $\mathfrak gl_{V}$ be the Lie algebra of endomorphisms on $V$. ...
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35 views

Subspace of a homogeneous space.

Given two homegenous spaces $\frac{G}{H}$, $\frac{A}{B}$ with $A\subset B$ is there a way to prove that $\frac{A}{B}\subset \frac{G}{H}$ ie that $B\subset A\cap H$? In particular I would like to ...
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1answer
466 views

Canonical isomorphism between $\mathfrak{so}(3)$ and $\mathbb R^3$ with vector cross product

There is a well-known isomorphism between the Lie algebra $\mathfrak{so}(3)$ and $\mathbb{R}^3$ which maps the Lie bracket to the vector cross product. It looks like $$ \begin{pmatrix} 0 & -z ...
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43 views

Solve commutator relation $[Q,d]=-[P,d]$ for $Q$ on chain complexes with scalar product

Suppose we are given chain sequences $\dots \rightarrow C_k \rightarrow C_{k+1} \rightarrow \dots$ and $\dots \rightarrow D_k \rightarrow D_{k+1} \rightarrow \dots$ of finite-dimensional vector ...
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2answers
519 views

Cayley Transform, Exponential Mapping and more…

Assume a self-adjoint operator, represented as hermitian matrix $H=H^\dagger$. To my knowledge there are at least 2 mappings of $H$ onto unitary matrices: Cayley's Transformation with ...
2
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1answer
77 views

Isomorphism between $sl_{4}$ and the orthogonal group of $6$ variables

Let V be the irreducible $sl_{4}$-module with highest weight $\pi_{2}=\lambda_{1}+\lambda_{2}$ (i.e if $H=\left(\lambda_{1},\dots,\lambda_{4}\right)$ is a diagonal matrix in $sl_{4}$ with values ...
2
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1answer
237 views

What physical meaning do the dimension of Wigner d-matrices have?

Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also ...
3
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1answer
52 views

$\hom_{k}\left(V_{p,q},V_{r,s}\right)\simeq V_{q+r,p+s}$

Define $V_{p,q}=\underset{p}{\underbrace{V\otimes\cdots\otimes V}}\otimes\underset{q}{\underbrace{V^{*}\otimes\cdots\otimes V^{*}}}$. In a previous question here I was shown that ...
5
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2answers
702 views

symplectic lie algebra is simple

The symplectic lie algebra defined by $sp\left(n\right)=\left\{ X\in gl_{2n}\,|\, X^{t}J+JX=0\right\}$ when $J=\begin{pmatrix}0 & I\\ -I & 0\end{pmatrix}$. So $X\in sp\left(n\right)$ is of ...